Cutting Planes Width and the Complexity of Graph Isomorphism Refutations
Article No.: 21, Pages 1 - 25
Abstract
The width complexity measure plays a central role in resolution and other propositional proof systems like Polynomial Calculus (under the name of degree). The study of width lower bounds is the most used method for proving size lower bounds, and it is known that for the mentioned proof systems, proofs with small width also imply the existence of proofs with small size. Not much has been studied, however, about the width parameter in the cutting planes (CP) proof system, a measure that was introduced by Dantchev and Martin in 2009 under the name of CP cutwidth.
In this article, we study the width complexity of CP refutations of graph isomorphism formulas. For a pair of non-isomorphic graphs \(G\) and \(H\), we show a direct connection between the Weisfeiler–Leman differentiation number \(\mathsf{WL}(G,H)\) of the graphs and the width of a CP refutation for the corresponding isomorphism formula \(\mathrm{Iso}(G,H)\). In particular, we show that if \(\mathsf{WL}(G,H)\leq k\), then there is a CP refutation of \(\mathrm{Iso}(G,H)\) with width \(k\), and if \(\mathsf{WL}(G,H) \gt k\), then there are no CP refutations of \(\mathrm{Iso}(G,H)\) with width \(k-2\). Similar results are known for other proof systems, like Resolution, Sherali–Adams, or Polynomial Calculus. We also obtain polynomial-length CP refutations from our width bound for isomorphism formulas for graphs with constant Weisfeiler–Leman dimension. Furthermore, we notice that a length lower bound for refuting graph isomorphism formulas in the subsystem of tree-like cutting planes with polynomially bounded coefficients follows from known results.
1 Introduction
Central to the field of combinatorial optimization is the \(\mathsf{NP}\)-hard problem of finding integer solutions to linear programs. This is done by optimizing the linear objective function \(\langle\mathbf{c},\mathbf{x}\rangle\) (for a given vector \({\mathbf{c}\in\mathbb{R}^{n}}\)) over the set of feasible points \(\mathbf{x}\) for the linear programming relaxation, described by a real polytope of the formwhere \(\mathbf{A}\in\mathbb{Z}^{m\times n}\) is some integer matrix, and \(\mathbf{b}\in\mathbb{Z}^{m}\) is an integer vector.1 If the polytope is integral (i.e., only contains integer vertices), one can optimize over all real vectors in \(P\) (i.e., solve the linear relaxation in polynomial time, for example, with the ellipsoid method of Khachiyan [42, 43]). Otherwise, one has to consider the integral hull \(P^{\mathbb{Z}}:=\operatorname{conv}(P\cap\mathbb{Z}^{n})\) for the optimization, i.e., the smallest polytope containing the integral points of \(P\). As was already suggested by Gomory [24] and later by Chvátal [13], in such a case, one can iteratively refine the set of feasible solutions by adding further valid constraints described by hyperplanes, or, more precisely, half-spaces, to the set of inequalities describing \(P\). These half spaces still contain \(P^{\mathbb{Z}}\) but—hopefully—cut off some parts of \(P\). For this purpose, the cut rule adds an inequality of the form \(\langle\mathbf{a},\mathbf{x}\rangle\geq\lceil b\rceil\) with an integral vector \(\mathbf{a}\) and a rational number \(b\) such that every point of \(P\) satisfies the inequality \(\langle\mathbf{a},\mathbf{x}\rangle\geq b\). If \(b\) is not an integer, then the former inequality, called Chvátal or Gomory–Chvátal (GC) inequality [13, 24], is not valid for (some) fractional solutions but still valid for all integer solutions. Intuitively, the hyperplane defined by the equality \(\langle\mathbf{a},\mathbf{x}\rangle=b\) is moved toward the polytope until an integer point (not necessarily of the polytope) is hit. This process yields a sequenceof polytopes. If some polytope \(P^{(i)}\) in this sequence is empty, \(P\) cannot have integer solutions.
\begin{align*}P=\{\mathbf{x}\in\mathbb{R}^{n}\mid\mathbf{A}\mathbf{x} \geq\mathbf{b},\ \mathbf{0}\leq\mathbf{x}\leq\mathbf{1}\},\end{align*}
\begin{align*}P\supseteq P^{(1)}\supseteq P^{(2)}\supseteq\dots\supseteq P^{ \mathbb{Z}},\end{align*}
Cutting Planes (CP) Proof System. Using this idea, Cook, Coullard, and Turán [17] introduced the CP proof system. In this system, one is initially given a set \(\big{\{}\sum_{j=1}^{n}a_{i,j}x_{j}\geq b_{i}\bigm{|}i\in[m]\big{\}}\) of integer inequalities describing the polytope \(P\). We also add the additional Boolean inequalities \(x_{j}\geq 0\) and \(-x_{j}\geq-1\) for each \(j\in[n]\), forcing these variables \(x_{j}\) to take values between \(0\) and \(1\) (this is a relaxation of the condition \(x_{j}\in\{0,1\}\)). Using the two deduction rules introduced in Definition 2.1 below, one can repeatedly deduce new inequalities, aiming to derive the contradictory inequality \(0\geq 1\). Obtaining a sequence of inequalities ending with \(0\geq 1\) is possible if and only if the initial set of inequalities does not admit an integer solution [17, 34]. This yields the CP proof system (formally introduced in Section 2.2).
In particular, CP can be used to refute unsatisfiable conjunctive normal form (CNF) formulas (by translating them into affine inequalities). CP is a strong proof system that can simulate Resolution, and it is exponentially stronger for several formula classes [17]. Exponential lower bounds on the length of a CP proof (as measured in the number of inequalities) have been shown using the interpolation method [31, 35, 52] and, more recently, using lifting and communication complexity results [22] that can be traced back to [10, 36, 39].
Other complexity measures for \(\mathrm{CP}\) have been studied. These measures are defined by the directed acyclic graph representing the proof (one connects the premises with the consequences). The rank of a proof is the maximum number of applications of the cut rule along any path in the directed graph. This is known as the Chvátal rank in linear optimization and was introduced in [11] in the area of proof complexity. This measure is the analogon of depth in Resolution [64]. Further, Dantchev and Martin [19] introduced the parameter cutwidth, defined as the maximum number of variables present in an inequality derived by performing a cut. This measure was further studied by Razborov [53] under the name of width, where the author presents linear lower bounds for this measure, as well as width/rank tradeoffs. In the case of Resolution, there is also a related complexity measure of width that measures how many literals are present in the largest clause in a refutation. In Polynomial Calculus, the analogous measure is degree. The seminal article [6] showed that proving width lower bounds for Resolution is a way to prove size lower bounds for Resolution. This result extends to the corresponding measures in Polynomial Calculus [14, 40]. These articles sparked interest in the width/degree complexity measures, resulting in a long line of articles proving lower bounds for these measures. The situation for \(\mathrm{CP}\) width lower bounds is dramatically more sparse. We are only aware of the two mentioned references [19, 53].
In this article, we study graph isomorphism (GI) formulas with respect to the parameters rank and width. This allows us to prove length upper bounds for isomorphism formulas based on graphs with constant Weisfeiler–Leman (WL) dimension. We also show lower bounds for these formulas in a subsystem of CP. A strong motivation for this study is that CP is a promising candidate to be used in future efficient implementations of satisfiability (SAT) solvers. Furthermore, proof complexity results also hold for all integer linear programming solvers based on the GC rule. These solvers provide up-to-date methods for solving \(\mathsf{NP}\)-hard Boolean optimization problems.
WL and Proof Complexity. The GI problem i.e., the task of deciding whether two given graphs are isomorphic, has been intensively studied and is well known for its unresolved complexity, as it is one of the few problems in \(\mathsf{NP}\) that is not known to be complete for this class nor to be in \(\mathsf{P}\). It is also unknown whether \(\mathrm{GI}\in\mathsf{co}\text{-}\mathsf{NP}\).
A naïve heuristic to distinguish two non-isomorphic graphs is the \(1\)-dimensional WL algorithm, or color refinement algorithm. This algorithm colors the vertices of the graphs according to their degree and then updates the original vertex colors according to the multiset of colors of their neighbors. This basic step is applied repeatedly until the colorings of the graphs stabilize. This procedure can be generalized to the \(k\)-dimensional WL algorithm (\(k\text{-}\mathrm{WL}\)) [65, 66]. In this more refined variant, the set of \(k\)-tuples of vertices is partitioned into automorphism-invariant equivalence classes (see, e.g., [44] for an overview of this procedure). It had been conjectured that \(\mathrm{GI}\) is solvable using the \(k\)-dimensional WL algorithm, with \(k\) being sublinear in the number of vertices of the graphs. However, this was shown to be false in the seminal work of Cai, Fürer, and Immerman [12]. Fascinatingly, the authors achieved this by relating the power of \(k\text{-}\mathrm{WL}\) to the expressive power of \(\mathscr{C}^{k}\), the \(k\)-variable fragment of first-order logic augmented with counting quantifiers, and a variant of an Ehrenfeucht-Fraïssé game [20, 21] called the bijective \(k\)-pebble game. Nevertheless, the WL method still plays a central role in the algorithmic research on \(\mathrm{GI}\); for example, Babai’s famous quasipolynomial time algorithm for \(\mathrm{GI}\) [3] uses the WL method as a subroutine.
The field of proof complexity provides a different approach to studying the complexity of the \(\mathrm{GI}\) problem. Here, one tries to find the smallest size of a proof of the fact that two graphs are non-isomorphic. It holds that \(\mathrm{GI}\) is in \(\mathsf{co}\text{-}\mathsf{NP}\) if and only if there is a concrete proof system with polynomial-size proofs of non-isomorphism. Similar to the Cook–Reckhow program [16] for the SAT problem, this defines a clear line of research trying to provide superpolynomial size lower bounds for refuting graph (non)isomorphism formulas in stronger and stronger proof systems. The situation is even more interesting here than in the \(\mathrm{SAT}\) case since it was proven in [5] that \(\mathrm{GI}\) is in \(\mathsf{co}\text{-}\mathsf{AM}\), a randomized version of \(\mathsf{co}\text{-}\mathsf{NP}\). Hence, it would not be too surprising if \(\mathrm{GI}\in\mathsf{co}\text{-}\mathsf{NP}\), and this would imply the existence of polynomial-size proofs for the problem in some system.
In a recent line of work, the power of different proof systems has been studied with respect to their power in refuting GI. The first example of such a lower bound was given in [61] for the Resolution proof system. This result led to lower bounds for stronger proof systems. These studies also make use of the WL algorithm. It has been shown in [1, 30, 49] that the power of \(k\text{-}\mathrm{WL}\) lies between the \(k\)th and \((k+1)\)-st level of the canonical Sherali–Adams linear programming hierarchy [58]. Moreover, the authors of [7] exactly characterized the power of the WL algorithm in terms of an algebraic proof system between degree-\(k\) Nullstellensatz and degree-\(k\) Polynomial Calculus. Furthermore, it was shown in [51] and independently in [15] that pairs of non-isomorphic \(n\)-vertex graphs exist such that any Sum-of-Squares proof of non-isomorphism must have degree \(\Omega(n)\). Closely related are the results of [2] that show that Sum-of-Squares degree and Polynomial Calculus degree correlate to the WL dimension (up to constant factors; we also refer to the very recent article [55]). Recently, in [62], an exact connection was shown between the width and depth measures in (narrow) resolution and the number of variables and the quantifier depth needed to distinguish a pair of graphs by first-order logic sentences. This result extends to a lower bound for the strong SRC-1 proof system, equipping Resolution with a symmetry rule [63].
1.1 Our Results and Techniques
We show a strong connection between the WL graph differentiation number and the geometric CP proof system. We write \(G\equiv_{k}^{\mathrm{CP}}H\) if there is no width-\(k\) CP refutation of \(\mathrm{Iso}(G,H)\), the set of inequalities encoding the statement that the graphs \(G\) and \(H\) are isomorphic. Further, we write \(G\equiv_{\mathscr{C}^{k}}H\) if the graphs \(G\) and \(H\) cannot be distinguished using the logic \(\mathscr{C}^{k}\), the \(k\)-variable fragment of first-order logic augmented with counting quantifiers. Our main result is the following theorem:
(Main Result). Let \(G\) and \(H\) be two non-isomorphic graphs. Then
\begin{align} G\equiv_{k}^{\mathrm{CP}}H\implies G\equiv_{\mathscr{C}^{k}}H \implies G\equiv_{k-2}^{\mathrm{CP}}H.\end{align}
(1)
In other words, if we let \(\mathsf{WL}(G,H)\) be the smallest \(k\) such that \(\mathscr{C}^{k}\) can distinguish the graphs \(G\) and \(H\), then:
(1)
If \(\mathsf{WL}(G,H)\leq k\) , then \(\mathrm{Iso}(G,H)\) can be refuted by CP using width \(k\).
(2)
If \(\mathsf{WL}(G,H) \gt k\) , then \(\mathrm{Iso}(G,H)\) is not refutable in CP using width \(k-2\).
We achieve the first result by using the winning positions of spoiler in the bijective pebble game to derive the necessary inequalities. The second result is shown by constructing a set of matrices that “protect” a given point in the isomorphism polytope from being cut away using cuts of a certain width. This result is achieved by proving a so-called protection lemma for GI. This type of lemmata has a long tradition in combinatorial optimization (see, e.g., [41]) and has also been used in the area of proof complexity in [11, 19, 47]. The concrete matrices are being constructed using winning positions for duplicator in the bijective pebble game. From the first result, we can derive polynomial-length \(\mathrm{CP}\) refutations for isomorphism formulas for graphs with constant WL dimension.
We also notice a length lower bound for refuting GI formulas in the subsystem of tree-like CP with polynomially bounded coefficients by using known results from communication complexity.
1.2 Organization of This Article
The remainder of this article is organized as follows. Section 2 introduces our notation, the CP proof system, the GC rule, our encoding of GI as a set of affine inequalities, and necessary tools from descriptive complexity. We proceed in Section 3 by showing the tight connection between the WL differentiation number for graphs and the width of refuting the corresponding GI formulas in the CP proof system. Section 4 establishes the lower bound for isomorphism formulas in Tree-CP with polynomially bounded coefficients.
2 Preliminaries
2.1 Notation
We let \(\mathbb{N}\) denote the set of positive integers and \(\mathbb{N}_{0}\) the set of non-negative integers. For \(n\in\mathbb{N}\), we define \([n]:=\{k\in\mathbb{N}\mid 1\leq k\leq n\}\). This article will denote tuples, vectors, and matrices in boldface. Given two vectors \(\mathbf{x},\mathbf{a}\in\mathbb{R}^{n}\), we let \(\langle\mathbf{a},\mathbf{x}\rangle:=\sum_{i=1}^{n}a_{i}x_{i}\) denote the standard dot product. If we have multiple vectors, say \(\mathbf{a}^{\mathbf{\boldsymbol{(}1\boldsymbol{)}}},\dots,\mathbf{a}^{\mathbf{ \boldsymbol{(}k\boldsymbol{)}}}\), we use superscripts. We reserve the subscript notation \(a_{i}\) for the \(i\) th coordinate of a vector \(\mathbf{a}=(a_{1},\dots,a_{n})\).
2.2 The CP Proof System
In this article, we consider CP as an inference system used for refuting unsatisfiable CNF formulas, as suggested by [17]. For this, a CNF formula \(F\) is translated into a system of affine inequalities that have a \(0\)-\(1\)-solution if and only if the corresponding assignment satisfies \(F\). These inequalities can then be manipulated according to certain rules. It is known that a formula is unsatisfiable if and only if, applying these rules, it is possible to obtain the contradiction \(0\geq 1\). A clause \(C=(\ell_{1}\lor\dots\lor\ell_{k})\) is converted to \(\tau(C)\equiv[\tau(\ell_{1})+\dots+\tau(\ell_{k})\geq 1]\), where for each literal \(\ell_{i}\), we let \(\tau(\ell_{i}):=x\) if \(\ell_{i}=x\) and \(\tau(\ell_{i}):=1-x\) if \(\ell_{i}=\neg x\). We also add the additional inequalities \(x\geq 0\) and \(-x\geq-1\) for each variable \(x\), forcing them to take values between \(0\) and \(1\) (this is a relaxation of the condition \(x\in\{0,1\}\)).
Let \(k\in\mathbb{N}\). Further, let \(\mathbf{a}\in\mathbb{Z}^{n}\), \(\mathbf{a}^{\mathbf{\boldsymbol{(}i\boldsymbol{)}}}\in\mathbb{Z}^{n}\), \(\gamma_{i}\in\mathbb{Z}\) for \(i\in[k]\), and \(\mathbf{x}=(x_{1},\dots,x_{n})\) be a vector of \(n\) variables. The proof system \(\mathrm{CP}\) has two derivation rules:
Linear combination:
From the inequalities \(\langle\mathbf{a}^{\mathbf{\boldsymbol{(}1\boldsymbol{)}}},\mathbf{x}\rangle \geq\gamma_{1},\dots,\langle\mathbf{a}^{\mathbf{\boldsymbol{(}k\boldsymbol{)}} },\mathbf{x}\rangle\geq\gamma_{k}\), and non-negative integers \(\alpha_{1},\dots,\alpha_{k}\in\mathbb{N}_{0}\), we can derive the inequality \(\sum_{i=1}^{k}\alpha_{i}\langle\mathbf{a}^{\mathbf{\boldsymbol{(}i\boldsymbol{) }}},\mathbf{x}\rangle\geq\sum_{i=1}^{k}\alpha_{i}\gamma_{i}\).
Rounding:
If all coefficients in the vector \(\mathbf{a}\) are divisible by a positive integer \(c\in\mathbb{N}\), we can derive the inequality \(\langle\frac{\mathbf{a}}{c},\mathbf{x}\rangle\geq\lceil\frac{\gamma}{c}\rceil\) from \(\langle\mathbf{a},\mathbf{x}\rangle\geq\gamma\).
While the linear combination rule is commonly formulated for \(k=2\) in the proof complexity literature, we have adopted the view of [11, 53], better reflecting linear programming. Let us emphasize that we allow arbitrary large values of \(k\) (i.e., \(k\) can even depend on \(n\)). We call the value \(k\) the fan-in of the proof.
As is standard (see, e.g., [41]), we will write \(a\leq b\) or \(-b\leq-a\) for an inequality of the form \(b\geq a\) when it is more natural in our arguments. If we have an inequality \(\langle\mathbf{a},\mathbf{x}\rangle\leq\gamma\) where all coordinates in \(\mathbf{a}\in\mathbb{Z}^{n}\) are divisible by some \(c\in\mathbb{N}\), the natural counterpart of the rounding rule allows the derivation of the inequality \(\langle\frac{\mathbf{a}}{c},\mathbf{x}\rangle\leq\lfloor\frac{\gamma}{c}\rfloor\), where the ceiling function is replaced with the floor function.
Without loss of generality, we can assume, as done in [11], that a rounding operation is always applied after each application of the linear combination rule and, therefore, both rules can be merged into a single one (called a GC cut in [11]).
A CP derivation of an inequality \(g\) from a set \(\mathscr{F}=\{f_{1},\dots,f_{m}\}\) of affine inequalities is a sequence \(\pi=(g_{1},\dots,g_{t})\) of affine inequalities such that
(1)
for each \(i\in[t]\), the inequality \(g_{i}\) is either an inequality from \(\mathscr{F}\) (an axiom) or is obtained from previous inequalities \(g_{j_{1}},\dots,g_{j_{k}}\) with \(j_{1} \lt \dots \lt j_{k} \lt i\) by a GC cut, and
(2)
\(g_{t}\) is the inequality \(g\).
If \(g\) is the inequality \(0\geq 1\), we speak of a CP refutation of \(\mathscr{F}\).
It is well-known that all the above-mentioned derivation rules are sound for integer solutions. Furthermore, the proof system is complete in the sense that each unsatisfiable CNF formula has a CP refutation (see, e.g., [13]).
A \(\mathrm{CP}\) refutation can be represented in the usual way as a directed acyclic graph in which each vertex corresponds to an affine inequality in the proof. The axioms are the sources, \(0\geq 1\) is the only sink, and for every application of a GC cut, there is an edge pointing from each of the vertices whose corresponding inequalities are involved in the cut to the vertex representing the result of the cut.
If we restrict the allowed proof structure to trees (i.e., every inequality, except for the axioms, is used at most once as a premise; otherwise, it has to be re-derived), we obtain the proof system \(\mathrm{Tree}\text{-}\mathrm{CP}\).
A family \((\mathscr{F}_{n})_{n\in\mathbb{N}}\) of sets of affine inequalities with \(\mathrm{O}(n)\) variables each has \(\mathrm{CP}\) refutations with polynomially bounded coefficients if
(1)
there exists a family \((\pi_{n})_{n\in\mathbb{N}}\) of \(\mathrm{CP}\) refutations for this family (i.e., for every \(n\in\mathbb{N}\), the proof \(\pi_{n}\) is a \(\mathrm{CP}\) refutation of \(\mathscr{F}_{n}\)), and
(2)
there exists a constant \(c \gt 0\) such that for all \(n\in\mathbb{N}\), the absolute value of all coefficients used in the inequalities of the refutation \(\pi_{n}\) is bounded by \(n^{c}\).
The system obtained by restricting CP to use polynomially bounded coefficients is denoted by \(\mathrm{CP}^{*}\). If we additionally require a tree-like proof structure, we get the system \(\mathrm{Tree}\text{-}\mathrm{CP}^{*}\).
The graphical representation of a \(\mathrm{CP}\) proof allows us to define some complexity measures. The most common complexity measure for a \(\mathrm{CP}\) refutation is its length, defined as the number of vertices in the refutation graph. Two other complexity measures play a central role in our results:
Let \(\pi=(g_{1},\dots,g_{t})\) be a \(\mathrm{CP}\) derivation of \(g_{t}\) from \(\mathscr{F}=\{f_{1},\dots,f_{m}\}\). We let \(\mathsf{Rank}(f_{i}):=0\) for all \(i\in[m]\). For any \(g\in\pi\) that was obtained by a GC cut from inequalities \(g_{j_{1}},\dots,g_{j_{k}}\) by dividing by \(c\), we let
\begin{align*}\mathsf{Rank}(g):=\left\{\begin{array}{@{}ll}\max\big\{\mathsf{Rank}(g_{j_{1}}),\dots,\mathsf{Rank}(g_{j_{k}})\big\} & \mbox{if }c=1,\\\max\big\{\mathsf{Rank}(g_{j_{1}}),\dots,\mathsf{Rank}(g_{j_{k}}) \big\}+1 & \mbox{if }c\neq 1.\end{array}\right.\end{align*}
The rank (also called depth) of \(\pi\) is defined as \(\mathsf{Rank}(g_{t})\).
This measure is known as the Chvátal rank in linear optimization (see [41, §19.6] for an excellent overview in this area) and was introduced in [11] in the area of proof complexity. It is the counterpart of depth in Resolution [64].
Dantchev and Martin [19] introduced the parameter cutwidth, defined as the maximum number of variables present in an inequality derived by performing a cut.
The cutwidth, or just width, of a \(\mathrm{CP}\) proof \(\pi\) is the maximum number of variables in an inequality that results from a GC cut involving rounding with an integer \(c\neq 1\). By this, we mean the number of variables remaining after the linear combination in the rule has been performed or, equivalently, the number of variables after the GC cut (linear combination plus rounding) has been done. If no GC cut with \(c\neq 1\) is used, we consider the cutwidth of \(\pi\) to be \(0\).
Finally, we agree on the following terminology.
Let \(\mathscr{Q}\in\{\mathrm{CP},\mathrm{CP}^{*},\mathrm{Tree}\text{-}\mathrm{CP}, \mathrm{Tree}\text{-}\mathrm{CP}^{*}\}\). For any complexity measure \(\mathsf{M}\) and any unsatisfiable system \(\mathscr{F}\) of affine inequalities, the \(\mathsf{M}\) -complexity of refuting \(\mathscr{F}\) in system \(\mathscr{Q}\) is the minimum value of \(\mathsf{M}(\pi)\) over all refutations \(\pi\) of \(\mathscr{F}\) in system \(\mathscr{Q}\).
2.3 Two Sets of Affine Inequalities for GI
We only deal undirected simple graphs. Such a graph is a tuple \(G=(V_{G},E_{G})\), where \(V_{G}\) is a finite set of vertices and \(E_{G}\subseteq\binom{V_{G}}{2}\) is the set of edges. For a vertex \(v\) in a graph \(G\), we denote by \(N_{G}(v)\) the set of its neighbors, and for a set of vertices \(S\), we define \(N_{G}(S)\) as the set of neighbors of the vertices in \(S\). If the graph is clear from the context, we drop the subscripts.
Two graphs \(G=(V_{G},E_{G})\) and \(H=(V_{H},E_{H})\) are isomorphic if there is a bijection \({\varphi\colon V_{G}\to V_{H}}\) (called isomorphism from \(G\) to \(H\)) such that \(\{u,v\}\in E_{G}\Leftrightarrow\big{\{}\varphi(u),\varphi(v)\big{\}}\in E_{H}\) holds for all \(u,v\in V_{G}\). We will denote this by \(G\cong H\).
Let \(G=(V_{G},E_{G})\) and \(H=(V_{H},E_{H})\) be two graphs with \(V_{G}=V_{H}=\{1,\dots,n\}\). We will use the set of variables \(x_{i,j}\) with \(i,j\in[n]\). If for an assignment \(\alpha\), we have \(\alpha(x_{i,j}) \gt 0\), this indicates that vertex \(i\) in \(G\) is mapped to vertex \(j\) in \(H\) under \(\alpha\).
For convenience, we consider two different sets of inequalities for which there is a satisfying integer assignment if and only if there is an isomorphism between \(G\) and \(H\).
2.3.1 Matrix Isomorphism Formulas.
The first set of affine inequalities is the one usually used in linear optimization. Let \(\mathbf{A}\) and \(\mathbf{B}\) be the adjacency matrices of the graphs \(G\) and \(H\). The graphs are isomorphic if and only if there is a permutation matrix \(\mathbf{X}\) satisfying
\begin{align}\mathbf{A}\mathbf{X}=\mathbf{X}\mathbf{B}.\end{align}
(2)
This relation lends itself to a linear programming relaxation: we relax the condition that \(\mathbf{X}\) is a permutation matrix to the condition that it is a doubly stochastic matrix.
A square matrix \(\mathbf{X}=(x_{i,j})\in\mathbb{R}^{n\times n}\) is called doubly stochastic if
(1)
\(0\leq x_{i,j}\leq 1\) for every \((i,j)\in[n]\times[n]\),
(2)
\(\sum_{j=1}^{n}x_{i,j}=1\) for every \(i\in[n]\), and
(3)
\(\sum_{i=1}^{n}x_{i,j}=1\) for every \(j\in[n]\).
Two graphs \(G\) and \(H\) are called fractional isomorphic if there exists a doubly stochastic matrix \(\mathbf{X}\) with \(\mathbf{A}\mathbf{X}=\mathbf{X}\mathbf{B}\), where \(\mathbf{A}\) and \(\mathbf{B}\) are the adjacency matrices of \(G\) and \(H\), respectively. The matrix \(\mathbf{X}\) is then called fractional isomorphism between \(G\) and \(H\).
In [59], it was shown that two graphs possess a fractional isomorphism if and only if the well-known color refinement algorithm (corresponding to equivalence in the logic \(\mathscr{C}^{2}\), see Definition 2.17) cannot distinguish them.
We now express the discussed relaxation of (2) by the following sets of inequalities, viewing the \((i,j)\)th entry of \(\mathbf{X}\) as the variable \(x_{i,j}\). To keep the following definition concise, we write two inequalities \(a\leq b\) and \(b\leq a\) as the equality \(a=b\).
Let \(G=(V_{G},E_{G})\) and \(H=(V_{H},E_{H})\) be two graphs. The set \(\mathrm{MIso}(G,H)\) (for matrix isomorphism) of inequalities contains the following axioms:
Type 1 axioms:
For every \(v\in V_{G}\), we include the equality \(\sum_{w\in V_{H}}x_{v,w}=1\); and for every \(w\in V_{H}\), we include the equality \(\sum_{v\in V_{G}}x_{v,w}=1\). These axioms mean that in the matrix \(\mathbf{X}\), the sum of each row, as well as the sum of each column, is one.
Type 2 axioms:
These encode the matrix product \(\mathbf{A}\mathbf{X}=\mathbf{X}\mathbf{B}\). For each position \((i,j)\in[n]\times[n]\), we have the equality \((\mathbf{A}\mathbf{X})_{i,j}=(\mathbf{X}\mathbf{B})_{i,j}\), or alternatively
\begin{align*}\sum_{k\in N_{G}(i)}x_{k,j}=\sum_{\ell\in N_{H}(j)}x_{i,\ell}.\end{align*}
Type 3 axioms:
For every variable \(x\), these are the usual Boolean \(\mathrm{CP}\) axioms \(x\geq 0\) and \(x\leq 1\).
2.3.2 Isomorphism Formulas.
An alternative set of affine inequalities over the same set of variables is sometimes more convenient and has been used before for encoding the isomorphism principle in other proof systems like Resolution [57, 61, 62] or Polynomial Calculus [7]. Instead of the inequalities for the matrices, for every two pairs of vertices \(v,v^{\prime}\in V_{G}\) and \(w,w^{\prime}\in V_{H}\) such that \(\{v,v^{\prime}\}\) is an edge in \(G\) and \(\{w,w^{\prime}\}\) is not an edge in \(H\) (or the other way around), we include an inequality, indicating that \(v\) is not mapped to \(w\) or \(v^{\prime}\) is not mapped to \(w^{\prime}\).
Let \(G=(V_{G},E_{G})\) and \(H=(V_{H},E_{H})\) be two graphs. The set \(\mathrm{Iso}(G,H)\) contains the following inequalities:
Type 1 and Type 3 axioms:
These are exactly the same as in \(\mathrm{MIso}(G,H)\).
Type 2 axioms:
For every \(v,v^{\prime}\in V_{G}\) and \(w,w^{\prime}\in V_{H}\) such that the mapping given by \(\big{\{}(v,w), (v^{\prime},w^{\prime})\big{\}}\) is not an isomorphism between the induced graphs \(G\big{[}\{v,v^{\prime}\}\big{]}\) and \(H\big{[}\{w,w^{\prime}\}\big{]}\), we include the inequality \(x_{v,w}+x_{v^{\prime},w^{\prime}}\leq 1\), indicating that an edge cannot be mapped to a non-edge or vice versa.
2.3.3 Comparison of Both Encodings.
Both systems of inequalities have the same set of \(0\)-\(1\) solutions, which encode the isomorphisms between \(G\) and \(H\) but can have different sets of fractional solutions. For example, setting all variables \(x_{i,j}\) to \(\frac{1}{n}\) is always a solution for \(\mathrm{Iso}(G,H)\) (even when the graphs are non-isomorphic) but only satisfies \(\mathrm{MIso}(G,H)\) when these are regular graphs.
We show that there are short \(\mathrm{CP}\) derivations of each set of inequalities from the other, although, for the derivation of \(\mathrm{MIso}(G,H)\) from \(\mathrm{Iso}(G,H)\), we need to use the GC-cut rule.
There is a polynomial-length \(\mathrm{CP}\) derivation of the set of inequalities \(\mathrm{Iso}(G,H)\) from \(\mathrm{MIso}(G,H)\) without using the GC cut rule (i.e., using only linear combinations).
We show how to derive a Type 2 inequality \(x_{i,j}+x_{k,\ell}\leq 1\) from \(\mathrm{MIso}(G,H)\) for the case in which \(\{i,k\}\in E_{G}\) but \(\{j,\ell\}\not\in E_{H}\). Starting from the Type 2 inequalitywe add the Type 1 axiom inequality \(-1\leq-\sum_{b\in[n]}x_{k,b}\), obtaining
\begin{align*}\sum_{a\in N_{G}(k)}x_{a,j}\leq\sum_{b\in N_{H}(j)}x_{k,b},\end{align*}
\begin{align*}\sum_{a\in N_{G}(k)}x_{a,j}-1\leq-\sum_{b\in\overline{N_{H}(j)}}x_{k,b}.\end{align*}
This last inequality can be rewritten as
\begin{align*}x_{i,j}+x_{k,\ell}+\sum_{a\in N_{G}(k)\setminus\{i\}}x_{a,j}+ \sum_{b\in\overline{N_{H}(j)}\setminus\{\ell\}}x_{k,b}\leq 1.\end{align*}
Adding the Type 3 axioms \(-x_{a,j}\leq 0\) and \(-x_{k,b}\leq 0\) for every \(a\in N_{G}(k)\setminus\{i\}\) and every \(b\in\overline{N_{H}(j)}\setminus\{\ell\}\), we obtain the desired inequality. ∎
Since fractional solutions can only be eliminated in \(\mathrm{CP}\) using the GC cut rule, this result implies that the set of solutions of \(\mathrm{MIso}(G,H)\) is included in the set of solutions of \(\mathrm{Iso}(G,H)\). We consider a derivation in the other direction:
For any two connected graphs \(G\) and \(H\) with maximum degree \(d\) and size at least \(2\), there is a polynomial-length \(\mathrm{CP}\) derivation with rank \(2\) and width \(2d\) of the set of inequalities \(\mathrm{MIso}(G,H)\) from \(\mathrm{Iso}(G,H)\).
For \(i,j\in[n]\), we show how to derive a Type 2 inequality like
\begin{align*}\sum_{a\in N_{G}(i)}x_{a,j}\geq\sum_{b\in N_{H}(j)}x_{i,b}.\end{align*}
The inequality in the other direction is completely analogous. Let some \(a\in\overline{N_{G}(i)}\) be fixed. Adding the axioms \(x_{a,j}+x_{i,b}\leq 1\) from \(\mathrm{Iso}(G,H)\) for each \(b\in N_{H}(j)\), we obtain
\begin{align}\big{|}N_{H}(j)\big{|}\cdot x_{a,j}+\sum_{b\in N_{H}(j)}x_{i,b} & \leq\big{|}N_{H}(j)\big{|}.\end{align}
(3)
From the axioms \(\sum_{b\in[n]}x_{i,b}\leq 1\) and \(-x_{i,b}\leq 0\), we can obtain \(\sum_{b\in N_{H}(j)}x_{i,b}\leq 1\), which multiplied times \(\big{|}N_{H}(j)\big{|}-1\) and added to Inequality (3) becomes
\begin{align}N_{H}(j)\big{|}\cdot x_{a,j}+\big{|}N_{H}(j)\big{|}\sum_{b \in N_{H}(j)}x_{i,b} & \leq 2\cdot\big{|}N_{H}(j)\big{|}-1.\end{align}
(4)
In the multiplication step, we have used that \(\big{|}N_{H}(j)\big{|}-1\geq 0\), which is true since the graphs are connected by assumption. Dividing Inequality (4) by \(\big{|}N_{H}(j)\big{|}\) and rounding, we obtain
\begin{align}+\sum_{b\in N_{H}(j)}x_{i,b} & \leq 1.\end{align}
(5)
This resulting inequality involves \(\big{|}N_{H}(j)\big{|}+1\) variables.
We can derive an inequality like (5) for each a \(\in\overline{N_{G}(i)}\). Adding them all, we obtain
\begin{align*}\sum_{a\in\overline{N_{G}(i)}}x_{a,j}+\big{|}\overline{N_{G}(i)} \big{|}\sum_{b\in N_{H}(j)}x_{i,b} & \leq\big{|}\overline{N_{G}(i)}\big{|}.\end{align*}
Together with the axiom \(-\sum_{a\in[n]}x_{a,j}\leq-1\), this becomes
\begin{align*}-\sum_{a\in{N_{G}(i)}}x_{a,j}+\big{|}\overline{N_{G}(i)}\big{|} \sum_{b\in N_{H}(j)}x_{i,b} & \leq\big{|}\overline{N_{G}(i)}\big{|}-1.\end{align*}
We can now add the axioms \(-x_{a,j}\leq 0\) for each \(a\in N_{G}(i)\), each of these inequalities multiplied times \(\big{|}\overline{N_{G}(i)}\big{|}-1\), reaching
\begin{align*}-\big{|}\overline{N_{G}(i)}\big{|}\sum_{a\in N_{G}(i)}x_{a,j}+ \big{|}\overline{N_{G}(i)}\big{|}\sum_{b\in N_{H}(j)}x_{i,b} & \leq\big{|}\overline{N_{G}(i)}\big{|}-1.\end{align*}
In the multiplication step, we have used the fact that \(\big{|}\overline{N_{G}(i)}\big{|}-1\geq 0\), which is true since \(i\not\in N_{G}(i)\). Dividing by \(\big{|}\overline{N_{G}(i)}\big{|}\) and rounding, we get
\begin{align*}\sum_{b\in N_{H}(j)}x_{i,b} & \leq\sum_{a\in N_{G}(i)}x_{a,j}.\end{align*}
In this last rounding step, there are \(\big{|}N_{G}(i)\big{|}+\big{|}N_{H}(j)\big{|}\) variables in the inequality. Sincethe result follows. ∎
\begin{align*} \max\{|N_{H}(j)|+1,|N_{G}(i)|+|N_{H}(j)|\} \leq 2d, \end{align*}
2.4 The WL Number and the Bijective k-Pebble Game
In order to express different properties of graphs by certain fragments of first-order logic sentences, Immerman introduced the following definition.
For a logic \(\mathscr{L}\) (of first-order logic sentences), we say that two graphs \(G\) and \(H\) are \(\mathscr{L}\) -equivalent, denoted by \(G\equiv_{\mathscr{L}}H\), if, for all sentences \(\psi\in\mathscr{L}\), it holds that
\begin{align*}G\vDash\psi\Longleftrightarrow H\vDash\psi.\end{align*}
Otherwise, we say that \(\mathscr{L}\) can distinguish \(G\) from \(H\), denoted by \(G\not\equiv_{\mathscr{L}}H\).
For \(p\in\mathbb{N}\), we introduce a counting quantifier \(\exists^{\geq p}\). The formula \(\exists^{\geq p}x \psi\) has the meaning that “there are at least \(p\) distinct \(x\) satisfying \(\psi\).” We also need the notion of quantifier depth (also called quantifier rank).
The quantifier depth of a first-order logic sentence \(\psi\) (possibly using counting quantifiers) is defined inductively as follows:
—
If \(\psi\) is atomic, then \(\operatorname{qd}(\psi):=0\);
—
\(\operatorname{qd}(\neg\psi):=\operatorname{qd}(\psi)\);
—
\(\operatorname{qd}(\psi_{1}\lor\psi_{2}):={\max}\big{\{}\!\operatorname{qd}(\psi_{1}), \operatorname{qd}(\psi_{2})\big{\}}\);
—
\(\operatorname{qd}(\exists^{\geq p}x \psi):=\operatorname{qd}(\psi)+1\).
The \(k\) -variable counting logic \(\mathscr{C}^{k}\) is the set of first-order logic formulas that use counting quantifiers but at most \(k\) different variables (possibly re-quantifying them). Further, \(\mathscr{C}^{k}_{r}\) is the subclass of \(\mathscr{C}^{k}\) where the quantifier depth in the formulas is restricted to \(r\).
For example, \(\exists x\big{[}\exists^{\geq 8}y E(x,y)\land\forall y\big{(}E(x,y)\to\exists ^{\geq 2}x E(y,x)\big{)}\big{]}\) lies in \(\mathscr{C}^{2}_{3}\) and says that there is a vertex that has at least 8 neighbors, each of which has at least two neighbors.
The WL differentiation number of two graphs \(G\) and \(H\) is defined by
\begin{align*}\mathsf{WL}(G,H):=\left\{\begin{array}{@{}ll}\min\{k \in\mathbb{N}\mid G\not\equiv_{\mathscr{C}^{k}}H\} & \mbox{if }G\not \cong H\\\infty & \mbox{if }G\cong H.\end{array}\right.\end{align*}
For a graph \(G\), we say that it has WL dimension at most \(k\) if and only if \(G\not\equiv_{\mathscr{C}^{k+1}}H\) for all graphs \(H\) non-isomorphic to \(G\).
Let \(G=(V_{G},E_{G})\) and \(H=(V_{H},E_{H})\) be two graphs for the remainder of this section. We describe the \(r\)-round bijective \(k\)-pebble game of Hella [33], adapting the excellent notation from [1]. This game can be used to test \(\mathscr{C}^{k}_{r}\)-equivalence. We first describe some notation and the concept of partial isomorphism before proceeding to introduce the game itself.
Let \(k\in\mathbb{N}\). Suppose \(\mathbf{v}=(v_{1},\dots,v_{k})\in(V_{G}\cup\{\star\})^{k}\). For \(i\in[k]\) and \(v\in V_{G}\cup\{\star\}\), we let \(\mathbf{v}[i/v]\) denote the tuple \((v_{1},\dots,v_{i-1},v,v_{i+1},\dots,v_{k})\). Further, we let \(|\mathbf{v}|_{\star}\) denote the number of stars in the tuple \(\mathbf{v}\).
Let \(k\in\mathbb{N}\) and let \(\mathbf{v}=(v_{1},\dots,v_{k})\in(V_{G}\cup\{\star\})^{k}\) and \(\mathbf{w}=(w_{1},\dots,w_{k})\in(V_{H}\cup\{\star\})^{k}\) be two \(k\)-tuples. We say that the pair \((\mathbf{v},\mathbf{w})\) induces/is a partial isomorphism2 between \(G\) and \(H\) if, for every \(i,j\in[k]\) we have:
(1)
\(v_{i}=\star\) if and only if \(w_{i}=\star\);
(2)
\(v_{i}=v_{j}\) if and only if \(w_{i}=w_{j}\);
(3)
\(\{v_{i},v_{j}\}\in E_{G}\) if and only if \(\{w_{i},w_{j}\}\in E_{H}\).
In the following game, Spoiler wants to exhibit a difference between the given graphs, while Duplicator tries to disguise such a difference by maintaining a partial isomorphism.
Let \(r\in\mathbb{N}_{0}\) and \(k\in\mathbb{N}\). The \(r\) -round bijective \(k\)-pebble game on the graphs \(G\) and \(H\) is played by two players, called Spoiler (he) and Duplicator (she). There are \(k\) pairs of matched pebbles in the game. The game proceeds in rounds. The game position after round \(r\) is finished can be represented by a pair \((\mathbf{v},\mathbf{w})\in\bigl{(}V_{G}\cup\{\star\}\bigr{)}^{k}\times\bigl{(} V_{H}\cup\{\star\}\bigr{)}^{k}\). The game starts with some initial position \((\mathbf{v}^{\mathbf{\boldsymbol{(}0\boldsymbol{)}}},\mathbf{w}^{\mathbf{ \boldsymbol{(}0\boldsymbol{)}}})\). If nothing else is stated, we always assume \(\mathbf{v}^{\mathbf{\boldsymbol{(}0\boldsymbol{)}}}=\mathbf{w}^{\mathbf{ \boldsymbol{(}0\boldsymbol{)}}}=(\star,\dots,\star)\). If the initial position does not induce a partial isomorphism between the graphs, Spoiler wins the game after \(0\) rounds. We now describe round \(r^{\prime}+1\) of the game. For this, we suppose that the position after round \(r^{\prime}\) is given by \((\mathbf{v},\mathbf{w})\).
—
Spoiler must choose an index \(i\in[k]\). If \(|\mathbf{v}|_{\star}=|\mathbf{w}|_{\star}=0\), the tuples are updated to \(\mathbf{v}[i/\star]\) and \(\mathbf{w}[i/\star]\) (otherwise, Spoiler can still opt to do this deletion step).
—
Duplicator then chooses a bijection \(\varphi\colon V_{G}\to V_{H}\). If no such bijection exists, she has lost.
—
Otherwise, Spoiler picks a vertex \(v\in V_{G}\) and an index \(j\in[k]\) such that \(v_{j}=w_{j}=\star\). The tuples are then updated to \(\mathbf{v}[j/v]\) and \(\mathbf{w}\big{[}j/\varphi(v)\big{]}\).
If the new \((\mathbf{v},\mathbf{w})\) does not induce a partial isomorphism, then Spoiler has won after \(r^{\prime}+1\) rounds. Otherwise, the game continues with the next round. We say that Duplicator has a winning strategy if she can make the game last indefinitely.
3 CP Refutations for Isomorphism Formulas
It is sometimes convenient to use an alternative view of the pebbling configurations used in Section 2.4.
Let \(G=(V_{G},E_{G})\) and \(H=(V_{H},E_{H})\) be two graphs. Further, let \(k\in\mathbb{N}\), \(\mathbf{v}=(v_{1},\dots,v_{k})\in\bigl{(}V_{G}\cup\{\star\}\bigr{)}^{k}\), and \(\mathbf{w}=(w_{1},\dots,w_{k})\in\bigl{(}V_{H}\cup\{\star\}\bigr{)}^{k}\). We writeto denote the set \(p\subseteq V_{G}\times V_{H}\) given by
\begin{align*}p=\operatorname{zip}(\mathbf{v},\mathbf{w})\end{align*}
\begin{align*}p:=\big{\{}(v_{i},w_{i})\bigm{|}i\in[k]\text{ such that }v_{ i}\neq\star\text{ and }w_{i}\neq\star\big{\}}.\end{align*}
Definition 2.20 can easily be adapted to game positions denoted in the way above.
Let \(G\) and \(H\) be two graphs. Further, let \(r\in\mathbb{N}_{0}\) and \(k\in\mathbb{N}\). For a game position \(p\subseteq V_{G}\times V_{H}\), we write \(p\in D^{k}(G,H)\) if \(p\) is a winning position for Duplicator in Hella’s bijective \(k\)-pebble game played on \(G\) and \(H\). Similarly, the set \(D^{k}_{r}(G,H)\) is defined to be the positions in which Duplicator does not lose in \(r\) rounds in the bijective \(k\)-pebble game. We use the notation \(S^{k}_{r}(G,H)\) to denote the set of winning positions for Spoiler in the respective game.
Let \(k\in\mathbb{N}\). As in [1, 7], we now define a relation on \(\bigcup_{\ell\leq k}\big{(}V_{G}^{\ell}\cup V_{H}^{\ell}\big{)}\).
Let \(k\in\mathbb{N}\) and \(\ell\leq k\). Further, let \(G=(V_{G},E_{G})\) and \(H=(V_{H},E_{H})\) be two graphs and let \(I,I^{\prime}\in\{G,H\}\), not necessarily distinct. Additionally, let \(\mathbf{u}=(u_{1},\dots,u_{\ell})\in V_{I}^{\ell}\) and \(\mathbf{u^{\prime}}=(u_{1}^{\prime},\dots,u_{\ell}^{\prime})\in V_{I^{\prime}} ^{\ell}\). We write \(\mathbf{u}\equiv_{D^{k}}\mathbf{u^{\prime}}\) if \(p:=\operatorname{zip}(\mathbf{u},\mathbf{u^{\prime}})=\big{\{}(u_{1},u_{1}^{ \prime}),\dots,(u_{\ell},u_{\ell}^{\prime})\big{\}}\in D^{k}(I,I^{\prime})\).
It was shown in [1, Lemma 3] that \(\equiv_{D^{k}}\) is an equivalence relation for every \(k \lt \mathsf{WL}(G,H)\).
Define the following two graphs:
\begin{align*}G:=\Big(\big\{v_{i}\bigm{|}i\in[4]\big\}, \big\{\{v_{1},v_{2}\}\big\} \Big)\quad\text{and}\quad H:=\Big(\big\{w_{j}\bigm{|}j \in[4]\big\}, \big\{\{w_{3},w_{4}\}\big\}\Big).\end{align*}
Let \(\mathbf{v}:=(v_{1},v_{2})\in V_{G}^{2}\). Then, \([\mathbf{v}]_{\equiv_{D^{2}}}=\big{\{}(v_{1},v_{2}), (v_{2},v_{1}), (w_{3},w _{4}), (w_{4},w_{3})\big{\}}\).
3.1 Constructing a CP Refutation from the Bijective Pebble Game
We show that if a pair \((G,H)\) of non-isomorphic graphs can be separated by the bijective \(k\)-pebble game in \(r\) rounds, then there is a \(\mathrm{CP}\) refutation for \(\mathrm{Iso}(G,H)\) having width \(k\) and rank \(r\) simultaneously. By Lemma 2.13, the same result holds for the \(\mathrm{MIso}(G,H)\) formulas. We use the equivalence relation \(\equiv_{D^{k}}\) to define a bipartite graph with certain properties.
Let \(G=(V_{G},E_{G})\) and \(H=(V_{H},E_{H})\) be two graphs. Further, let \(r\in\mathbb{N}_{0}\) and \(k\in\mathbb{N}\). Moreover, let \(p\subseteq V_{G}\times V_{H}\) be an initial position of the bijective \(k\)-pebble game played on the graphs \(G\) and \(H\). We define the bipartite graph \(B:=B^{k}_{r}(p):=\big{(}V_{G}\uplus V_{H},E_{B}\big{)}\) with the edge set
\begin{align*}E_{B}:=\left\{\{v,w\} \middle| p\cup\big{\{}(v,w)\big{\}}\not\in S^{k}_{r}(G,H)\right\}.\end{align*}
We need the following result from Berkholz and Grohe:
Let \(G=(V_{G},E_{G})\) and \(H=(V_{H},E_{H})\) be two non-isomorphic graphs. Further, let \(p\subseteq V_{G}\times V_{H}\). Suppose that Spoiler has a winning strategy in the bijective \(k\)-pebble game played on the graphs \(G\) and \(H\) in \(r+1\) rounds starting from position \(p\). Then, in the graph \(B:=B^{k}_{r}(p)\) there are two sets \(S\subseteq V_{G}\) and \(T\subseteq V_{H}\) with the following properties:
—
\(N_{B}(S)=T\) , \(N_{B}(T)=S\) , and \(|S| \gt |T|\) ;
—
Spoiler can win the bijective \(k\) -pebble game in \(r\) rounds from the starting position \(p\cup\big{\{}(v,w)\big{\}}\) for every pair \((v,w)\in V_{G}\times V_{H}\) with the property \(v\in S\Leftrightarrow w\not\in T\).
By assumption, \(p\in S^{k}_{r+1}(G,H)\). This means that for all possible bijections \(\varphi\colon V_{G}\to V_{H}\) that Duplicator can provide, there is always a \(v\in V_{G}\) that Spoiler can choose in return, such that he still has a winning strategy in the bijective \(k\)-pebble game from the position \(p\cup\big{\{}\big{(}v,\varphi(v)\big{)}\big{\}}\) in \(r\) rounds. Hence for this \(v\), we have \(\big{\{}v,\varphi(v)\big{\}}\not\in E_{B}\). Thus, there can be no perfect matching in the graph \(B\). By Hall’s marriage theorem [32], a set \(S\subseteq V_{G}\) exists with \(\big{|}N_{B}(S)\big{|} \lt |S|\). We choose \(S\) to be an inclusion-maximal set with this property. Further, let
\begin{align*}T:=N_{B}(S).\end{align*}
We claim that \(N_{B}(T)=S\) holds. To reach a contradiction, suppose that there is a vertex
\begin{align} v\in N_{B}(T)\setminus S.\end{align}
(6)
The maximality of \(S\) implies that \(N_{B}(v)\not\subseteq T\). Letbe a vertex witnessing this fact. Since \(v\in N_{B}(T)\), there exists a vertex
\begin{align} w\in N_{B}(v)\setminus T,\end{align}
(7)
\begin{align} w^{\prime}\in N_{B}(v)\cap T.\end{align}
(8)
Moreover, since \(T=N_{B}(S)\), there is a vertex
\begin{align} v^{\prime}\in N_{B}(w^{\prime})\cap S.\end{align}
(9)
The choice of the vertices in Equations (6)–(9) implies that there are edges
\begin{align*}\{v^{\prime},w^{\prime}\}, \{v,w^{\prime }\}, \{v,w\}\in E_{B}.\end{align*}
This means that Duplicator has a winning strategy for the \(r\)-round bijective \(k\)-pebble game from the starting positions \(p\cup\big{\{}(v^{\prime},w^{\prime})\big{\}}\), \(p\cup\big{\{}(v,w^{\prime})\big{\}}\), and \(p\cup\big{\{}(v,w)\big{\}}\). Since \(\equiv_{D^{k}}\) is an equivalence relation, we thus have that she also has a winning strategy starting from the position \(p\cup\big{\{}(v^{\prime},w)\big{\}}\). Hence, \(\{v^{\prime},w\}\in E_{B}\). However, this contradicts \(w\not\in N_{B}(S)\). ∎
For a game position \(p=\big{\{}(v_{1},w_{1}),\dots,(v_{\ell},w_{\ell})\big{\}}\subseteq V_{G}\times V _{H}\) we let \(S_{p}:=\sum_{i=1}^{\ell}x_{v_{i},w_{i}}\). Note that, in particular, \(S_{\emptyset}=0\).
Let \(G=(V_{G},E_{G})\) and \(H=(V_{H},E_{H})\) be two non-isomorphic graphs with \(|V_{G}|=|V_{H}|\). Further, let \(p_{0}\subseteq V_{G}\times V_{H}\). Suppose that Spoiler has a winning strategy for the \(r\)-round bijective \(k\)-pebble game played on the graphs \(G\) and \(H\) starting from the initial position \(p_{0}\). Then, there is a CP derivation of the inequality
\begin{align*}S_{p_{0}}\leq|p_{0}|-1,\end{align*}
from \(\mathrm{Iso}(G,H)\) having width \(k\) and rank \(r\) simultaneously.
We prove the theorem by induction on \(r\), the number of rounds in the game. First, we consider the base case, where Spoiler wins the game from \(p_{0}\) in \(0\) rounds. Since \(\big{|}V_{G}\big{|}=\big{|}V_{H}\big{|}\), it must be that \(p_{0}\) is not a local isomorphism; therefore, there are two pairs \((v,w),(v^{\prime},w^{\prime})\in p_{0}\) that induce a local non-isomorphism. Hence, the inequality \(x_{v,w}+x_{v^{\prime},w^{\prime}}\leq 1\) must be a Type 2 axiom of \(\mathrm{Iso}(G,H)\). Adding the Type 3 axiom inequalities \(x_{a,b}\leq 1\) for all \(|p_{0}|-2\) many other pairs \((a,b)\in p_{0}\setminus\big{\{}(v,w), (v^{\prime},w^{\prime})\big{\}}\), we obtain a derivation of \(S_{p_{0}}\leq|p_{0}|-1\). This derivation has rank \(0\), as it is a linear combination of axioms. Furthermore, its width is upper bounded by \(|p_{0}|\leq k\).
For the induction step, let \(p\subseteq p_{0}\) with \(|p|=:\ell \lt k\) be the set of pairs not deleted by Spoiler at the beginning of the first round in the game. Assume that Spoiler has a winning strategy with \(k\) pebble pairs in \(r+1\) rounds from \(p\) and that the induction hypothesis holds. It thus suffices to show that it is possible to derive the inequality \({S_{p}\leq|p|-1}\) from \(\mathrm{Iso}(G,H)\) in width \(k\) and rank \(r+1\) simultaneously. For this purpose, we consider the bipartite graph \(B:=B^{k}_{r}(p)\) from Definition 3.5. Since \(p\in S^{k}_{r+1}(G,H)\), we know from Lemma 3.6 that there are two sets \(S\subseteq V_{G}\) and \(T\subseteq V_{H}\) with \(N_{B}(S)=T\), \(N_{B}(T)=S\), and \(|S| \gt |T|\), and such that for every pair \((v,w)\) with \(v\in S\Leftrightarrow w\not\in T\), Spoiler can win the game with \(k\) pebble pairs in \(r\)-rounds from the start position \(p\cup\big{\{}(v,w)\big{\}}\). By the induction hypothesis, there is a CP derivation of \(S_{p}+x_{v,w}\leq|p|\) in both width \(k\) and rank \(r\) for all such pairs.
We notice first that we can derive the inequalities
\begin{align}\sum_{v\in S, w\in T}x_{v,w}\leq|T|\quad\text{ and }\quad\sum_{v \in\overline{S}, w\in\overline{T}}x_{v,w}\leq|\overline{S}|.\end{align}
(10)
To derive the first one of them, observe that for each \(w\in T\), we have the axiom inequality \(\sum_{v\in V_{G}}x_{v,w}\leq 1\), which can be reduced to \(\sum_{v\in S}x_{v,w}\leq 1\) by adding the axioms \(x_{v,w}\geq 0\) for all \(v\in\overline{S}\). We obtain the first expression by adding the inequalities for all \(w\in T\). The second one is completely analogous. Adding both inequalities of (10) together, we get
\begin{align}\sum_{v\in S, w\in T}x_{v,w}+\sum_{v\in\overline{S}, w\in\overline{T}}x_{v,w }\leq|T|+|\overline{S}|.\end{align}
(11)
Since \(|T|+|\overline{S}| \lt |S|+|\overline{S}|=n\), Inequality (11) can be weakened toby first adding up the Boolean axioms \(x\leq 1\) and \(-x\leq 0\) to obtain \(0\leq 1\), and then by multiplying this inequality by the appropriate constant and adding it to (11).
\begin{align}\sum_{v\in S, w\in T}x_{v,w}+\sum_{v\in\overline{S}, w\in\overline{T}}x_{v,w }\leq n-1.\end{align}
(12)
Next, for each vertex \(v\in S\), adding over all inequalities \(S_{p}+x_{v,w}\leq|p|=\ell\) corresponding to pairs \((v,w)\) with \(w\in\overline{T}\) (derived inductively), we getand adding over all \(v\in S\), we obtain
\begin{align*}\sum_{w\in\overline{T}}(S_{p}+x_{v,w})\leq|\overline{T}|\ell,\end{align*}
\begin{align}\sum_{v\in S}\sum_{w\in\overline{T}}(S_{p}+x_{v,w})\leq|S||\overline{T}|\ell.\end{align}
(13)
Analogously, considering the pairs \((v,w)\) with \(v\not\in S\) and \(w\in T\), we get
\begin{align}\sum_{w\in T}\sum_{v\in\overline{S}}(S_{p}+x_{v,w})\leq|\overline{S}||{T}|\ell.\end{align}
(14)
Let \(\gamma:=|S||\overline{T}|+|\overline{S}||{T}|\). By adding the inequalities corresponding to the long Type 1 axioms for all vertices \(v\in V_{G}\), we can derive the inequality
\begin{align*}\sum_{v\in V_{G}, w\in V_{H}}x_{v,w}\geq n.\end{align*}
Subtracting (13) and (14) from this, we get
\begin{align*}\sum_{v\in S, w\in T}x_{v,w}+\sum_{v\in\overline{S}, w\in\overline{T}}x_{v,w }-\gamma S_{p}\geq n-\gamma\ell.\end{align*}
By subtracting (12), we derive
\begin{align*}-\gamma S_{p}\geq 1-\gamma\ell.\end{align*}
Observe that this last inequality has been obtained as the linear combination of axioms and previous inequalities, and therefore, the derivation can be done in one step. Using the rounding rule dividing by \(\gamma\), we getwhich is equivalent to \(S_{p}\leq\ell-1\). The linear combination and the rounding rule count as one use of the GC rule (using rank \(1\) and width \(|p|\)). ∎
\begin{align*}-S_{p}\geq\left\lceil\frac{1-\gamma\ell}{\gamma}\right\rceil=1-\ell,\end{align*}
Let \(G\) and \(H\) be two non-isomorphic graphs with \(|V_{G}|=|V_{H}|\). If \({G\not\equiv_{\mathscr{C}^{k}_{r}}H}\), then there is a CP refutation of \(\mathrm{Iso}(G,H)\) having width \(k\) and rank \(r\) simultaneously.
Since \({G\not\equiv_{\mathscr{C}^{k}_{r}}H}\), Spoiler can win the bijective \(k\)-pebble game in \(r\) rounds starting from the empty initial position \(p_{0}=\emptyset\). Since \(S_{\emptyset}=0\), the above result implies that the contradiction \(0\leq-1\) can be derived with the desired parameters. ∎
Let us give two applications of this corollary for the rank measure:
(1)
Grohe and Kiefer [28] have shown that there is a \(k\) such that \(\mathscr{C}^{k}\) identifies every \(n\)-vertex planar graph using a quantifier depth bounded by \(\mathrm{O}(\log n)\) (i.e., distinguishes it from every other non-isomorphic graph). This means that a CP rank of \(\mathrm{O}(\log n)\) suffices to prove non-isomorphism for planar graphs.
(2)
For random graphs, color refinement asymptotically almost surely terminates after \(2\) iterations [4] (i.e., the fraction of graphs with \(n\) vertices for which this is not true tends to \(0\) as \(n\to\infty\)). This means that CP asymptotically almost surely needs only rank \(2\) to prove the non-isomorphism of random graphs.
Even more excitingly, we can also get length upper bounds for CP refutations of GI, using Theorem 3.7 and Corollary 3.8.
If a pair \((G,H)\) of non-isomorphic graphs with \(n\) vertices each can be separated by the bijective \(k\)-pebble game, then there is a CP refutation of \(\mathrm{Iso}(G,H)\) having length \(n^{\mathrm{O}(k)}\).
The claim follows from the observation that the CP refutation of \(\mathrm{Iso}(G,H)\) constructed in Theorem 3.7 only contains linear combinations of axioms or inequalities of the form \(S_{p}\leq|p|-1\) for game positions \(p\subseteq V_{G}\times V_{H}\) with \(|p|\leq k\). Since there are at most \(\sum_{i=0}^{k}\binom{n}{i}^{2}=n^{\mathrm{O}(k)}\) such sets of pairs, the result follows. ∎
Grohe [27, Corollary 18.4.1] proved that two non-isomorphic graphs in every graph class characterized by a non-empty set of forbidden minors can be distinguished using \(\mathscr{C}^{k}\) for a constant \(k\). This implies that for these graphs with excluded minors, the CP procedure can produce polynomial-size proofs of graph non-isomorphism (the size of the coefficients in the used inequalities is polynomial in \(n\) since they are all either of the form \(S_{p}\leq|p|-1\) or linear combinations of axioms or such inequalities). As a concrete example, we mention that it was shown in [45] that the WL dimension of the class of all finite planar graphs is at most \(3\).
As mentioned above, the article [4] shows that almost all (finite) graphs are characterized by \(\mathscr{C}^{2}\) (i.e., the fraction of graphs with \(n\) vertices that are not identified by color refinement tends to \(0\) as \(n\to\infty\)). It is well-known that this result fails for regular graphs. However, \(\mathscr{C}^{3}\) asymptotically almost surely decides isomorphism for random regular graphs [9, 46]. This means that given two non-isomorphic graphs (maybe even random regular), CP asymptotically almost surely can produce polynomial-size certificates of graph non-isomorphism for the input graphs.
3.2 CP Width Lower Bound for Isomorphism Formulas
As described in [13, 41], for a polytope \(P\subset\mathbb{R}^{n}\), the Chvátal closure \(P^{\prime}\) is the polytope defined bythat is, we remove all points of the polytope \(P\) that are (in a certain sense) definitely not integer solutions. By iteratively defining \(P^{(i+1)}:=(P^{(i)})^{\prime}\), we obtain a sequence \(P=P^{(0)}\supseteq P^{(1)}\supseteq P^{(2)}{\supseteq\dots}\) of polytopes. The Chvátal rank can then be seen as the smallest \(r\) such that \(P^{(r)}=P^{\mathbb{Z}}\) (it was shown by Schrijver [56] that such an \(r\) always exists).
\begin{align} P^{\prime}:=\left\{\mathbf{x}\in P \middle| \begin{array}{ll}\forall\mathbf{a}\in\mathbb{Z}^{n},\ \forall b\in\mathbb{R}: \\\big{[}\forall\mathbf{y}\in P:\langle\mathbf{a},\mathbf{y}\rangle\geq b\big{]} \implies\langle\mathbf{a},\mathbf{x}\rangle\geq\lceil b\rceil\end{array}\right \};\end{align}
(15)
Protection lemmata have a long tradition in optimization theory for the study of the Chvátal rank. For the \(\mathrm{CP}\) rank, such lemmata have been used in [11] and [47]. A protection lemma for \(\mathrm{CP}\) width was introduced in [19]. We give a width protection lemma adapted to the GI problem. This generalizes (15). The following notation is employed.
Given a matrix \(\mathbf{X}\in\mathbb{R}^{n\times n}\) and a set \(I\subseteq[n]\) we denote by \(\mathbf{X}|_{I}\in\mathbb{R}^{n\times n}\) the projection of \(\mathbf{X}\) to the rows in \(I\), that is, to the positions \(\textrm{Rows}[I]:=\{(i,j)\mid i\in I, j\in[n]\}\) (meaning that the rows which are not in \(I\) are set to 0).
Let \(G\) and \(H\) be two graphs with \(n\) vertices each and let \(P_{G,H}\) be the polytope in \([0,1]^{n\times n}\) defined by the \(\mathrm{MIso}(G,H)\) inequalities. For \(k\in\mathbb{N}\), we define
\begin{align*}P^{\prime}_{G,H}(k):=\left\{\mathbf{X}\in P_{G,H} \middle|\begin{array}{ll@{}}\forall\mathbf{A}\in\mathbb{Z}^{n\times n},\ \forall b\in\mathbb{R},\ \forall I\subseteq[n]\text{ with }|I|=k:\\\big{[}\forall \mathbf{Y}\in P_{G,H}:\langle\mathbf{A},\mathbf{Y}|_{I}\rangle _{\text{F}}\geq b\big{]}\implies\langle\mathbf{A},\mathbf{X}|_{I}\rangle _{\text{F}}\geq\lceil b\rceil\end{array}\right\}.\end{align*}
Here \(\langle\mathbf{A},\mathbf{B}\rangle_{\text{F}}:=\sum_{i=1}^{n}\sum_{j=1} ^{n}a_{i,j} b_{i,j}\) denotes the Frobenius inner product between the matrices \(\mathbf{A}=(a_{i,j})\in\mathbb{R}^{n\times n}\) and \(\mathbf{B}=(b_{i,j})\in\mathbb{R}^{n\times n}\).
We move on to the protection lemma for GI. Intuitively, the idea of this lemma is that if a fractional isomorphism has an associated set of “protection matrices” in the GI polytope, then this set protects the fractional isomorphism from being cut away when moving from \(P_{G,H}\) to \(P^{\prime}_{G,H}(k)\).
Let \(G\) and \(H\) be two graphs with \(n\) vertices each. Additionally, let \(k\in\mathbb{N}\). Further, let \(\mathbf{X}\in[0,1]^{n\times n}\) be a fractional isomorphism in the polytope \(P_{G,H}\). Suppose that for any \(I\subseteq[n]\) with \(|I|\leq k\), there are matrices \(\mathbf{Y}^{\mathbf{\boldsymbol{(}1\boldsymbol{)}}},\dots,\mathbf{Y}^{\mathbf{ \boldsymbol{(}s\boldsymbol{)}}}\in[0,1]^{n\times n}\) satisfying:
—
For all \(t\in[s]\) , \((\mathbf{Y}^{\mathbf{\boldsymbol{(}t\boldsymbol{)}}})_{i,j}\in\{0,1\}\) in all positions \((i,j)\in\mathrm{Rows}[I]\);
—
for all \(t\in[s]\) , the matrix \(\mathbf{Y}^{\mathbf{\boldsymbol{(}t\boldsymbol{)}}}\) is a fractional solution of \(P_{G,H}\) ; and
—
the restriction \(\mathbf{X}|_{I}\) is a convex combination of \(\mathbf{Y}^{\mathbf{\boldsymbol{(}1\boldsymbol{)}}}|_{I},\dots,\mathbf{Y}^{ \mathbf{\boldsymbol{(}s\boldsymbol{)}}}|_{I}\).
Then, \(\mathbf{X}\in P^{\prime}_{G,H}(k)\).
Suppose, to reach a contradiction, that \(\mathbf{X}\) is a fractional isomorphism in \(P_{G,H}\) but \(\mathbf{X}\not\in P^{\prime}_{G,H}(k)\). Then, by Definition 3.11 there exists a matrix \(\mathbf{A}\in\mathbb{Z}^{n\times n}\), a real number \(b\in\mathbb{R}\), and a set \(I\subseteq[n]\) with \(|I|=k\) such that for all \(\mathbf{Y}\in P_{G,H}\) we have \(\langle\mathbf{A},\mathbf{Y}|_{I}\rangle_{\text{F}}\geq b\) but \(\langle\mathbf{A},\mathbf{X}|_{I}\rangle_{\text{F}} \lt \lceil b\rceil\). Since \(\mathbf{X}\in P_{G,H}\), we have \(\langle\mathbf{A},\mathbf{X}|_{I}\rangle_{\text{F}}\geq b\).
For all the protection matrices \(\mathbf{Y}^{\mathbf{\boldsymbol{(}t\boldsymbol{)}}}\in\big{\{}\mathbf{Y}^{ \mathbf{\boldsymbol{(}1\boldsymbol{)}}},\dots,\mathbf{Y}^{\mathbf{\boldsymbol{ (}s\boldsymbol{)}}}\big{\}}\), since they are \(0\)-\(1\)-valued in \(\mathrm{Rows}[I]\), we have that \(\langle\mathbf{A},\mathbf{Y}^{\mathbf{\boldsymbol{(}t\boldsymbol{)}}}|_{I} \rangle_{\text{F}}\) is an integer. Also, since \(\mathbf{Y}^{\mathbf{\boldsymbol{(}t\boldsymbol{)}}}\) is in the polytope, \(\langle\mathbf{A},\mathbf{Y}^{\mathbf{\boldsymbol{(}t\boldsymbol{)}}}|_{I} \rangle_{\text{F}}\geq b\). Combining both facts, we have \(\langle\mathbf{A},\mathbf{Y}^{\mathbf{\boldsymbol{(}t\boldsymbol{)}}}|_{I} \rangle_{\text{F}}\geq\lceil b\rceil\). However, since \(\mathbf{X}|_{I}\) is a convex combination of the protection matrices, it must hold \(\langle\mathbf{A},\mathbf{X}|_{I}\rangle_{\text{F}}\geq\lceil b\rceil\), which is a contradiction. ∎
Note that for \(|I|=k\), the above restrictions consider \(kn\) variables. In previously published protection lemmata, these restrictions had size \(k\). However, our version can only make the construction of the protection matrices harder.
For each game position \(p\subseteq V_{G}\times V_{H}\), we define a matrix \(\mathbf{M^{p}}\) that we will show in Lemma 3.22 to be a fractional isomorphism between \(G\) and \(H\). We begin by first defining auxiliary functions that will be used to define the entries of this matrix. Since \(\equiv_{D^{k}}\) is an equivalence relation, we can define the type of a tuple as the equivalence class of it.
Let \(G=(V_{G},E_{G})\) and \(H=(V_{H},E_{H})\) be two graphs and let \(I,I^{\prime}\in\{G,H\}\), not necessarily distinct. Additionally, let \(k\in\mathbb{N}\) and \(\ell\leq k\). Given a tuple \(\mathbf{v}\in V_{I}^{\ell}\), we letbe the equivalence class of \(\mathbf{v}\) with respect to the \(k\)-pebble game played on \((I,I^{\prime})\). Further, we define
\begin{align*}c_{k}(\mathbf{v}):=[\mathbf{v}]_{\equiv_{D^{k}}}\end{align*}
\begin{align*}t_{k}(\mathbf{v}):=\left|c_{k}(\mathbf{v})\cap V_{I}^{ |\mathbf{v}|}\right|.\end{align*}
Before moving on, let us discuss several examples and simple properties of these two notions, as they are central to the rest of our arguments.
If \(c_{k}(\mathbf{v})=c_{k}(\mathbf{v^{\boldsymbol{\prime}}})\), then \((\mathbf{v},\mathbf{v^{\boldsymbol{\prime}}})\) clearly is a partial isomorphism. The converse is not true.
It is straightforward to verify that the value of the function \(t_{k}\) is invariant under permutations of the tuple \(\mathbf{v}\).
To define the matrix \(\mathbf{M^{p}}\), we will use the following auxiliary function.
Let \(G=(V_{G},E_{G})\) and \(H=(V_{H},E_{H})\) be two graphs. Additionally, let \(k\in\mathbb{N}_{0}\) and \(\ell\leq k\). Further, let \(\mathbf{v}\in V_{G}^{\ell}\) and \(\mathbf{w}\in V_{H}^{\ell}\). For every non-empty game position \({q:=\operatorname{zip}(\mathbf{v},\mathbf{w})\neq\emptyset}\), the function \(\zeta_{k}\) is defined in the following way:
\begin{align}\zeta_{k}(q):=\left\{\begin{array}{ll}0 & \mbox {if }c_{k}(\mathbf{v})\neq c_{k}(\mathbf{w}),\\\frac{1}{t_{k}(\mathbf{v})} & \mbox{otherwise}.\end{array}\right.\end{align}
(16)
For \(q=\emptyset\), we let \(\zeta_{k}(\emptyset):=1\).
Given a game position \(p\) in the \(k\)-pebble game, we now use the function \(\zeta_{k}\) to define the entries \(m^{p}_{i,j}\) of the matrix \(\mathbf{M^{p}}\). For a game position \(p\subseteq V_{G}\times V_{H}\) and a tuple \((v,w)\in V_{G}\times V_{H}\), we use the notation \(p\cup vw\) as a shorthand for \(p\cup\big{\{}(v,w)\big{\}}\).
Let \(G=(V_{G},E_{G})\) and \(H=(V_{H},E_{H})\) be two graphs. Additionally, let \(k\in\mathbb{N}\). Further, let \(p\subseteq V_{G}\times V_{H}\) be a game position in the bijective \(k\)-pebble game with \(|p|\leq k-1\). For every \(i,j\in[n]\), the number \(m^{p}_{i,j}\) is defined in the following way:
\begin{align} m^{p}_{i,j}:=\left\{\begin{array}{ll}0 & \mbox {if }p\cup v_{i}w_{j}\not\in D^{k}(G,H),\\\frac{\zeta_{k}(p\cup v_{i}w_{j})}{\zeta_{k}(p)} & \mbox{otherwise}.\end{array}\right.\end{align}
(17)
We further define the matrix \(\mathbf{M^{p}}\) by letting \((\mathbf{M^{p}})_{i,j}:=m^{p}_{i,j}\) for every \(i,j\in[n]\).
Observe that in the case \(\zeta_{k}(p)=0\), the value of \(m^{p}_{i,j}\) is \(0\) because the first case in (16) implies that \(p\cup v_{i}w_{j}\not\in D^{k}(G,H)\), ensuring that we do not divide by zero in (17). The following result follows directly from the definition of the matrix entries.
Let \(G\) and \(H\) be two graphs. Further, let \(k\in\mathbb{N}\). If \(p\in D^{k}(G,H)\) and \((v_{i},w_{j})\in p\), then
(1)
\(m^{p}_{i,j}=1\) , and
(2)
\(m^{p}_{i^{\prime},j}=m^{p}_{i,j^{\prime}}=0\) for every \(i^{\prime}\neq i\) and every \(j^{\prime}\neq j\).
Consider the graphs in Figure 1 in the 2-bijective pebble game. Let \(p:=\big{\{}(1,1)\big{\}}\). We claim that then
\begin{align*}\mathbf{M^{p}}=\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0\\0 & \frac{1}{2} & 0 & 0 & 0 & \frac{1}{2}\\0 & 0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0\\0 & 0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0\\0 & 0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0\\0 & \frac{1}{2} & 0 & 0 & 0 & \frac{1}{2}\\\end{pmatrix}.\end{align*}
Fig. 1.
Let us demonstrate the calculation of some entries of the corresponding matrix, in order to make the above definitions more accessible:
—
Lemma 3.18 immediately specifies the entries of the first row and column of the matrix.
—
Clearly, \(m^{p}_{2,3}=0\) by Definition 3.17 since \(p\cup\big{\{}(2,3)\big{\}}=\big{\{}(1,1), (2,3)\big{\}}\not\in D^{2}(G,H)\).
—
Let us now demonstrate how \(m^{p}_{2,2}\) is calculated. Since \(p\cup\big{\{}(2,2)\big{\}}=\big{\{}(1,1), (2,2)\big{\}}\in D^{2}(G,H)\), we havesince \(t_{2}(1)=\big{|}\{1,\dots,6\}\big{|}\) and \(t_{2}(1,2)=\left|\big{\{}(i,j)\bigm{|}\{i,j\}\in E_{G}\big{\}}\right|=2\cdot 6\).
\begin{align*}m^{p}_{2,2}=\frac{\zeta_{2}(p\cup v_{i}w_{j})}{\zeta_{2}(p)}=\frac{t_{2}(1)}{t _{2}(1,2)}=\frac{6}{12}=\frac{1}{2},\end{align*}
—
Arguing less formally, \(m^{p}_{3,3}=\frac{1}{3}\), since \(\{1,3\}\not\in E_{G}\) and there are \(9\) such non-edges \(\{1,3\}\), \(\{1,4\}\), \(\{1,5\}\), \(\{2,4\}\), \(\{2,5\}\), \(\{2,6\}\), \(\{3,5\}\), \(\{3,6\}\), and \(\{4,6\}\) in \(G\).
Let \(\mathbf{v}=(v_{1},\dots,v_{k})\in V_{G}^{k}\). For \(v\in V_{G}\), we let \(\mathbf{v}v:=(v_{1},\dots,v_{k},v)\).
In the following, we tacitly assume an ordering \(v_{1}\prec v_{2}\prec\dots\prec v_{n}\) on the vertices of the graph and often identify a vertex \(v_{i}\) with its number \(i\) in this order. Hence, we can now speak of matrix positions \((v,w)\). This helps to keep the following notation clear. The following technical lemma is needed in the next results. It follows from the properties of the equivalence relations defined by the bijective pebble game.
Let \(G=(V_{G},E_{G})\) and \(H=(V_{H},E_{H})\) be two graphs. Additionally, let \(k\in\mathbb{N}\). Moreover, let \(p:=\operatorname{zip}(\mathbf{a},\mathbf{b})\) with \(|p|\leq k-1\), where \(\mathbf{a}\in V_{G}^{|p|}\) and \(\mathbf{b}\in V_{H}^{|p|}\). Then, for every \((v,w)\in V_{G}\times V_{H}\), if \(c_{k}(\mathbf{a}v)=c_{k}(\mathbf{b}w)\), then
\begin{align*}t_{k}(\mathbf{a} v)=t_{k}(\mathbf{a})\cdot\left|{\{}w^ {\prime}\in V_{H}{|}c_{k}(\mathbf{a} v)=c_{k}(\mathbf{b} w^{\prime})\right|.\end{align*}
Let \((v,w)\in V_{G}\times V_{H}\) with \(c_{k}(\mathbf{a}v)=c_{k}(\mathbf{b}w)\). This equality implies
\begin{align*}\left|{\{}w^{\prime}\in V_{H}{|}c_{k}(\mathbf{a} v)=c_{k}(\mathbf{b} w^{\prime}){\}}\right|=\left|{ \{}v^{\prime}\in V_{G}{|}c_{k}(\mathbf{a} v)=c_{k}(\mathbf{a} v^{\prime}){\}}\right|.\end{align*}
Also, from the fact that for every \(\mathbf{a^{\prime}}\in c_{k}(\mathbf{a})\), there are \(\left|\big{\{}v^{\prime}\in V_{G}\bigm{|}c_{k}(\mathbf{a}v)=c_{k}(\mathbf{a}v^ {\prime})\big{\}}\right|\) many extensions \(\mathbf{a^{\prime}}v^{\prime}\) such that \(c_{k}(\mathbf{a^{\prime}}v^{\prime})=c_{k}(\mathbf{a}v)\), we have
\begin{align*}\left|\big{\{}v^{\prime}\in V_{G}\bigm{|}c_{k}(\mathbf{a} v)=c_{k}(\mathbf{a} v^{\prime})\big{\}}\right|=\frac{\left |c_{k}(\mathbf{a} v)\right|}{\left|c_{k}(\mathbf{a})\right|}=\frac {t_{k}(\mathbf{a} v)}{t_{k}(\mathbf{a})}.\end{align*}
∎
Let \(G=(V_{G},E_{G})\) and \(H=(V_{H},E_{H})\) be two graphs. Further, let \(k\in\mathbb{N}\setminus\{1\}\). Moreover, let \(p:=\operatorname{zip}(\mathbf{a},\mathbf{b})\) with \(|p| \lt k-1\), where \(\mathbf{a}\in V_{G}^{|p|}\) and \(\mathbf{b}\in V_{H}^{|p|}\) are tuples such that \(c_{k}(\mathbf{a})=c_{k}(\mathbf{b})\). Then, the matrix \(\mathbf{M^{p}}\) is a fractional isomorphism between \(G\) and \(H\).
The definition of the matrix immediately implies that the Type 3 axioms are satisfied. We next show that when \(p\) is as above, the axioms expressing that we are dealing with a double stochastic matrix are satisfied by \(\mathbf{M^{p}}\) (even when \(|p|=k-1)\). For each \(v\in V_{G}\), we have
\begin{align*}\sum_{w\in V_{H}}(\mathbf{M^{p}})_{v,w}=\sum_{\begin{subarray}{c}w\in V_{H}\\p\cup vw\in D^{k}(G,H)\end{subarray}}\frac{\zeta_{k}(p\cup vw)}{\zeta_{k}(p)}= \sum_{\begin{subarray}{c}w\in V_{H}\\c_{k}(\mathbf{a} v)=c_{k}(\mathbf{b} w)\end{subarray}} \frac{t_{k}(\mathbf{a})}{t_{k}(\mathbf{a} v)}.\end{align*}
Since we are adding over the \(w\in V_{H}\) with \(c_{k}(\mathbf{a}v)=c_{k}(\mathbf{b}w)\), by Lemma 3.21, we have
\begin{align*}\left|\big\{w\in V_{H}\bigm{|}c_{k}(\mathbf{a} \nu)=c_{k}(\mathbf{b} w)\big\}\right|=\frac{t_{k}(\mathbf{a}\nu)}{t_{k}(\mathbf{a})}.\end{align*}
Hence, the sum of a row in the matrix adds to 1. The proof for the columns is analogous.
For the case of the isomorphism axioms, let \((v,w)\in V_{G}\times V_{H}\). We have to show the equality
\begin{align}\sum_{i\in N_{G}(v)}m^{p}_{i,w}=\sum_{j\in N_{H}(w)}m^{p}_{v,j}.\end{align}
(18)
By the result on the double stochasticity of the matrices just proved above, since \(\left|{p\cup iw}\right|\leq k-1\), we have that for every \(i\in N_{G}(v)\),where the last equality holds because, according to Definition 3.16, only for the neighbors \(j\) of \(w\) the value of \(m^{p\cup iw}_{v,j}\) can be different from \(0\). This is true since, because of the assumption \(i\in N_{G}(v)\), we can have \(c_{k}(\mathbf{a}iv)=c_{k}(\mathbf{b}wj)\) only for \(j\in N_{H}(w)\). By definition, if \(m^{p\cup iw}_{v,j}\neq 0\), then
\begin{align*}1=\sum_{j\in V_{H}}m^{p\cup iw}_{v,j}=\sum_{j\in N_{H}(w)}m^{p\cup iw}_{v,j},\end{align*}
\begin{align*}m^{p\cup iw}_{v,j}=\frac{\zeta_{k}(p\cup iw\cup vj)}{\zeta_{k}(p\cup iw)}= \frac{t_{k}(\mathbf{a} i)}{t_{k}(\mathbf{a} i v)}.\end{align*}
Similarly, if \(m^{p}_{v,j}\neq 0\), then
\begin{align*}m^{p}_{v,j}=\frac{t_{k}(\mathbf{a})}{t_{k}(\mathbf{a} v)}.\end{align*}
Hence, if \(m^{p\cup iw}_{v,j}\neq 0\), this number can be expressed as \(\xi\cdot m^{p}_{v,j}\), with
\begin{align*}\xi:=\frac{t_{k}(\mathbf{a} v)\cdot t_{k}(\mathbf{a} i)}{t_{k}(\mathbf{a})\cdot t_{k}(\mathbf{a} i v)}.\end{align*}
Therefore,
\begin{align*}\sum_{j\in N_{H}(w)}m^{p}_{v,j}=\frac{1}{\xi}\sum_{j\in N_{H}(w)}m^{p\cup iw}_ {v,j}=\frac{1}{\xi}.\end{align*}
Similarly, for every \(j\in N_{H}(w)\), we haveand likewise, the numbers \(m^{p\cup vj}_{i,w}\) (when different from \(0\)) can be expressed as the product \(\xi\cdot m^{p}_{i,w}\). Therefore, both sums in (18) are equal. ∎
\begin{align*}1=\sum_{i\in N_{G}(v)}m^{p\cup vj}_{i,w},\end{align*}
We observe that in the previous result, it does not suffice that \(|p|\leq k-1\) in order for the matrix \(\mathbf{M^{p}}\) to be a fractional isomorphism between \(G\) and \(H\). As a counterexample consider the graphs \(G\) and \(H\) in Figure 1, and let \(\mathbf{A}\) and \(\mathbf{B}\) be the adjacency matrices of these graphs. Duplicator wins the 2-pebble game on \((G,H)\), however, it can be easily checked that for \({p}=(v,w)\) for any pair \((v,w)\in V_{G}\times V_{H}\), the matrix \(\mathbf{M^{p}}\)—given in Example 3.19 for the case \(p=\big{\{}(1,1)\big{\}}\)—does not satisfy \(\mathbf{A}\mathbf{M^{p}}=\mathbf{M^{p}}\mathbf{B}.\)
Let \(G\) and \(H\) be two non-isomorphic graphs with \(n\) vertices each such that \(G\equiv_{\mathscr{C}^{k}}H\). Further, let \(p\in D^{k}(G,H)\) with \(|p| \lt k-1\) and consider the matrix \(\mathbf{M^{p}}\). For any \(I\subseteq[n]\) with \(|I| \lt k-1\), there is a set \(\big{\{}\mathbf{Y}^{\mathbf{\boldsymbol{(}1\boldsymbol{)}}},\dots,\mathbf{Y}^{ \mathbf{\boldsymbol{(}s\boldsymbol{)}}}\big{\}}\) of matrices satisfying:
—
Each of these matrices is \(0\) - \(1\) -valued on \(\mathrm{Rows}[I]\);
—
each of these matrices is a fractional isomorphism in \(P_{G,H}\) of the form \(\mathbf{M^{p^{\boldsymbol{\prime}}}}\) , with \(|p^{\prime}| \lt k-1\) and \(p^{\prime}\in D^{k}(G,H)\) ; and
—
the restriction \(\mathbf{M^{p}}|_{I}\) is a convex combination of \(\mathbf{Y}^{\mathbf{\boldsymbol{(}1\boldsymbol{)}}}|_{I},\dots,\mathbf{Y}^{ \mathbf{\boldsymbol{(}s\boldsymbol{)}}}|_{I}\).
We prove the result by induction on \(\ell=|I|\). For the induction base \(\ell=1\), let \(p\) be a game starting position as above and let \(I=\{i_{1}\}\). Consider the matrix \(\mathbf{M^{p}}\). We can, without loss of generality, suppose that \(i_{1}\) is not a vertex contained in a tuple of \(p\) and also that for all \(j\in[n]\), the matrix entry \(m^{p}_{i_{1},j}\neq 1\). Otherwise, \(\mathbf{M^{p}}\) already has the desired properties.
Let us start with the simpler case in which \(|p| \lt k-2\). Note that not all the entries in row \(i_{1}\) of \(\mathbf{M^{p}}\) can be \(0\) since we are dealing with a fractional isomorphism according to Lemma 3.22. Under these conditions, there is a set of at least two non-zero elements in that row; let us call this set \(\hbox{NZ}(i_{1})\). This follows from the fact that the sum of the row elements adds to \(1\). For each \(j\in\hbox{NZ}(i_{1})\), letand consider the matrices \({\mathbf{Y}^{\mathbf{\boldsymbol{(}j\boldsymbol{)}}}:=\mathbf{M^{p_{j}}}}\) for \(j\in\hbox{NZ}(i_{1})\). Notice that the game position \(p_{j}\) can be reached from \(p\) in one round of the bijective \(k\)-pebble game.
\begin{align} p_{j}:=p\cup v_{i_{1}}w_{j},\end{align}
(19)
These matrices have the following properties: According to Lemma 3.18, they have \(0\)-\(1\) values in row \(i_{1}\). Due to Lemma 3.22, the matrices \(\mathbf{M^{p_{j}}}\) are fractional isomorphisms since \(|p_{j}| \lt k-1\), and all the \(p_{j}\) considered as game positions are winning positions for Duplicator in the bijective \(k\)-pebble game since they can be reached if Spoiler adds the pairs \((v_{i_{1}},w_{j})\) for any \(j\in\hbox{NZ}(i_{1})\), which each are valid moves since these positions are non-zero in \(\mathbf{M^{p}}\), meaning that \(p\cup v_{i_{1}}w_{j}\) is also a winning position for Duplicator. It is only left to show that \(\mathbf{M^{p}}|_{I}\) is a convex combination of the restriction to \(\mathrm{Rows}[I]\) of the matrices \(\mathbf{M^{p_{j}}}\), but this follows from the fact that for each \(j\in\hbox{NZ}(i_{1})\), the matrix \(\mathbf{M^{p_{j}}}\) has a \(1\) in position \((i_{1},j)\) and \(0\)’s in all other positions in the row \(i_{1}\), and all these positions have the same value in \(\mathbf{M^{p}}\). Therefore, \(\mathbf{M^{p}}|_{I}\) can be obtained as a convex combination of the restriction to \(\mathrm{Rows}[I]\) of the new matrices, multiplying each of them times the number \((\mathbf{M^{p}})_{i_{1},j}\). This is a correct combination since for \(p=\operatorname{zip}(\mathbf{a},\mathbf{b})\) and \(c_{k}(\mathbf{a}v_{i_{1}})=c_{k}(\mathbf{b}w_{j})\), we haveand the number of such matrices is equal to
\begin{align*}(\mathbf{M^{p}})_{i_{1},j}=\frac{t_{k}(\mathbf{a})}{t_{k}(\mathbf{a} v_{i_{1}})},\end{align*}
\begin{align*}\big{|}\hbox{NZ}(i_{1})\big{|}=\left|\big\{j\in[n]\bigm{|}c_{k}(\mathbf{a} v_{i_{1}})=c_{k}(\mathbf{b} w_{j})\big\} \right|=\frac{t_{k}(\mathbf{a} v_{i_{1}})}{t_{k}(\mathbf{ a})}.\end{align*}
Let us now suppose \(|p|=k-2\). In this situation, we cannot just add elements to \(p\) since then, we cannot guarantee that the resulting matrix is a fractional isomorphism, and we have to delete some elements from \(p\) first. Let \((v,w)\) be any pair in \(p\) and let
\begin{align}\hat{p}:=p\setminus\big\{(v,w)\big\}.\end{align}
(20)
For any \(j\in\hbox{NZ}(i_{1})\), we letand consider the matrices \({\mathbf{Y}^{\mathbf{\boldsymbol{(}j\boldsymbol{)}}}:=\mathbf{M^{p^{ \boldsymbol{\prime}}_{j}}}}\) (notice that the game position \(p^{\prime}_{j}\) can be reached from \(p\) in just one round of the bijective \(k\)-pebble game). Again, these matrices have \(0\)-\(1\) values in row \(i_{1}\) and encode fractional isomorphisms since each \(p^{\prime}_{j}\) has the right cardinality and is a winning position for Duplicator in the bijective \(k\)-pebble game. This is true since these positions can be reached if Spoiler deletes \((v,w)\) from \(p\) and adds \((v_{i_{1}},w_{j})\), which are valid moves since these positions are non-zero in \(\mathbf{M^{p}}\), meaning that \(p\cup v_{i_{1}}w_{j}\) is also a winning position for Duplicator and, therefore, \(p^{\prime}_{j}\) is all the more one. It is only left to show that \(\mathbf{M^{p}}|_{I}\) is a convex combination of the restriction to \(\mathrm{Rows}[I]\) of the matrices \(\mathbf{M^{p^{\boldsymbol{\prime}}_{j}}}\). Let \(p=\operatorname{zip}(\mathbf{a}v,\mathbf{b}w)\). The value of a non-zero position in row \(i_{1}\) in \(\mathbf{M^{p}}\) is given by
\begin{align} p^{\prime}_{j}:=\hat{p}\cup v_{i_{1}}w_{j},\end{align}
(21)
\begin{align*}\frac{t_{k}(\mathbf{a} v)}{t_{k}(\mathbf{a} v v_{i_{1}})}.\end{align*}
If there is a non-zero position in row \(i_{1}\) in \(\mathbf{M^{p}}\), then the same position in \(\mathbf{M}^{\boldsymbol{\hat{}}{\mathbf{p}}}\) is also non-zero since \(\hat{p}\subseteq p\). Each matrix \(\mathbf{M^{p^{\boldsymbol{\prime}}_{j}}}\) has a \(1\) in position \((i_{1},j)\), and \(0\)’s in the other positions in that row. If \((\mathbf{M^{p}})_{i_{1},j}\neq 0\), then \(\mathbf{M^{p^{\boldsymbol{\prime}}_{j}}}\) is one of the \(\mathbf{Y}\) matrices since \((\mathbf{M}^{\boldsymbol{\hat{}}{\mathbf{p}}})_{i_{1},j}\neq 0\). There arenon-zero positions in row \(i_{1}\) in \(\mathbf{M^{p}}\). Multiplying the \(\mathbf{Y}\) matrices corresponding to these positions times \(\frac{t_{k}(\mathbf{a}v)}{t_{k}(\mathbf{a}vv_{i_{1}})}\) and adding them together, we obtain the convex combination.
\begin{align*}\left|\big{\{}j\in[n]\bigm{|}c_{k}(\mathbf{a} v\mskip 2.0 mu v_{i_{1}})=c_{k}(\mathbf{b} w w_{j})\big{\}}\right|=\frac{t_{k}(\mathbf{a} v v_{i_{1}})}{t_{k}(\mathbf{a} v)}\end{align*}
The induction step is completely analogous. Given \(\mathbf{M^{p}}\) and \(I=\{i_{1},\dots,i_{\ell}\}\), we let \(I^{\prime}:=\{i_{1},\dots,i_{\ell-1}\}\). By the induction hypothesis, we can construct a set of matrices \(\mathbf{Y}^{\mathbf{\boldsymbol{(}t\boldsymbol{)}}}\) of the form \(\mathbf{M^{p^{\boldsymbol{\prime}}}}\) with \(p^{\prime}\) containing a set of pairs \(\big{\{}(i_{1},j_{1}),\dots,(i_{\ell-1},j_{\ell-1})\big{\}}\) satisfying the conditions and such that \(\mathbf{M^{p}}|_{I^{\prime}}\) is a convex combination of the constructed matrices. These are \(0\)-\(1\) on \(\mathrm{Rows}[I^{\prime}]\). In one last step, we can construct from these the matrices for \(I\) as in the case of \(\ell=1\). A convex combination of convex combinations is still one. ∎
If Duplicator has a winning strategy for the bijective \(k\)-pebble game played on the graphs \(G\) and \(H\), then there is no \(\mathrm{CP}\) refutation of \(\mathrm{MIso}(G,H)\) of width \(k-2\).
This follows from Lemma 3.12 and the previous result since they together imply that each \(\mathbf{M^{p}}\) corresponding to a winning position \(p\) for Duplicator of size \(|p|\leq k-2\) survives cuts of size \(k-2\). At each step, starting from the empty position \(p=\emptyset\) we consider the fractional isomorphism \(\mathbf{M^{p}}\). There are protection matrices for it that also correspond to winning positions \(p^{\prime}\) for the Duplicator with size \(|p^{\prime}|\leq k-2\). For each of these new positions \(p^{\prime}\) there are also protection matrices and therefore, it is not possible, allowing only cuts of width \(k-2\), to eliminate any of these fractional isomorphisms from \(P_{G,H}\). ∎
A close inspection of the proof of the previous theorem, together with Lemma 3.12, also gives a connection to \(\mathrm{CP}\) rank.
Let \(k\geq 3\). If Duplicator has a winning strategy for the \(r\)-round bijective \(k\)-pebble game played on the graphs \(G\) and \(H\), then there are no \(\mathrm{CP}\) refutations of \(\mathrm{MIso}(G,H)\) of width \(k-2\) and rank \(\left\lfloor\frac{r}{k-2}\right\rfloor\).
The hypothesis implies that the empty position belongs to \(D^{k}_{r}(G,H)\). For any \(r^{\prime}\leq r\) and any position \(p\in D^{k}_{r^{\prime}}(G,H)\), the protection matrices \(\mathbf{M^{p^{\boldsymbol{\prime}}}}\) for \(\mathbf{M^{p}}\) correspond to positions \(p^{\prime}\) with \(|p^{\prime}|\leq k-2\) that can be reached from \(p\) in the bijective \(k\)-pebble game in at most \(k-2\) rounds (the maximum number of round happens when \(p\cap p^{\prime}=\emptyset)\). This is because, in the proof of Theorem 3.23, the modification of the pebble configuration described in Equation (19) needs one round. The same is true for the combined modifications described in Equations (20) and (21). This means that the new position \(p^{\prime}\) belongs to \(D^{k}_{r^{\prime}-(k-2)}(G,H)\). Therefore, if the cuts have a width of at most \(k-2\) in a \(\mathrm{CP}\) refutation of \(\mathrm{MIso}(G,H)\), the rank has to be at least \(\left\lfloor\frac{r}{k-2}\right\rfloor\). ∎
4 Tree-CP \({}^{\ast}\) Length Lower Bounds for Refuting Isomorphism
Proving length lower bounds for CP refutations of the isomorphism problem is a challenging open question. Basically, the two only known methods for proving such bounds are interpolation and lifting. Neither of these methods is suitable for isomorphism formulas. Interpolation requires some monotone problem, and \(\mathrm{GI}\) is highly non-monotone. Also, after applying lifting, one obtains some constructed formulas that are not isomorphism formulas. Using some known results from communication complexity, we can, however, show length lower bounds for the restricted case of tree-like CP proofs with polynomially bounded coefficients (we denote this system with Tree-CP\({}^{*}\)) and constant fan-in. The system Tree-CP\({}^{*}\) (with constant fan-in) is non-trivial, allowing, for example, polynomial-length proofs for the pigeonhole principle [17].
In [39], the length of Tree-CP proofs (with constant fan-in) for a formula \(F\) was related to the communication complexity of a search problem for \(F\), showing that if the underlying search problem has high communication complexity, this implies a lower bound for the Tree-CP\({}^{*}\) length of refuting \(F\).
Critical block sensitivity is a communication complexity measure introduced in [36], extending the classical concept of block sensitivity [50]. It is an easy fact that a critical block sensitivity lower bound for a problem implies the same bound for the communication complexity of the search problem.
In [26, Theorem 3], lower bounds on the critical block sensitivity of Tseitin formulas were proved. The authors showed that there exist graph families of bounded degree, with critical block sensitivity communication \(\Omega(n/\log n)\) which by the results in [39] imply a length lower bound of \(2^{\Omega(n/\log^{2}n)}\) for Tree-CP\({}^{*}\) refutations of Tseitin formulas.
It was shown in [60, Lemma 4.2] that there is a direct reduction from Tseitin to isomorphism formulas, and it is, thus, possible to obtain lower bounds for isomorphism formulas from lower bounds for Tseitin formulas. As a direct consequence of all these results, we obtain:
There are families of non-isomorphic graphs \(G_{n}\not\cong H_{n}\) with \(n\) vertices each, such that any refutation of \(\mathrm{MIso}(G_{n},H_{n})\) in \(\mathrm{Tree}\text{-}\mathrm{CP}^{*}\) (with constant fan-in) requires length \(2^{\Omega(n/\log^{2}n)}\).
The second author’s Ph.D. dissertation [67] provides more details on this observation.
5 Conclusions and Open Problems
We have shown a characterization of the WL graph differentiation number of two graphs in terms of the cutwidth needed for refuting the corresponding isomorphism formula. Let us emphasize that Equation (1) holds for both the Iso and MIso formulas. For this, we have introduced a new protection lemma for the GI polytope. This new connection enabled us to show that the CP proof system can show graph non-isomorphism in polynomial time for graphs with a constant WL dimension. Furthermore, by using known results from communication complexity, we were able to give a lower bound for the length of tree-like \(\mathrm{CP}\) refutations with polynomially bounded coefficients for refuting GI inequalities. Some important questions remain open. Maybe the most interesting one is to prove \(\mathrm{CP}\) length lower bounds for isomorphism formulas. This is quite challenging since basically all the lower bounds for this kind of formula are based on graphs related to the Tseitin formulas, and recently a quasipolynomial upper bound for the \(\mathrm{CP}\) length of such formulas has been shown [18]. Furthermore, it would be interesting to have tradeoff results between the dimension of the WL algorithm and its iteration number (this is equivalent to a tradeoff between the number \(k\) of pebbles in Hella’s bijective pebble game and the number \(r\) of rounds). While tradeoff results are known for these parameters [8, 29], they do not hold for structures of bounded arity (like graphs). However, due to the connection of these parameters to the Resolution proof system [62] and the CP proof system, as shown in this article, such results would immediately imply proof complexity tradeoffs (in our case, between width and rank for CP). Moreover, it is open if the second implication in (1) can be improved.
Footnotes
1
Note that we restrict our attention to polytopes in \([0,1]^{n}\) rather than polyhedrons in \(\mathbb{R}^{n}\).
2
We also refer to this as a local isomorphism.
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Index Terms
- Cutting Planes Width and the Complexity of Graph Isomorphism Refutations
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Publication History
Published: 29 October 2024
Online AM: 18 July 2024
Accepted: 27 June 2024
Revised: 21 April 2024
Received: 20 November 2023
Published in TOCL Volume 25, Issue 4
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