We provide several advances to the understanding of the class of quantum Merlin--Arthur (QMA) proof systems, the quantum analogue of NP. Our central contribution is proving a longstanding conjecture that the consistency of local density matrices (CLDM) ...

This paper considers a special class of nonlocal games $(G,\psi)$, where $G$ is a two-player one-round game, and $\psi$ is a bipartite state independent of $G$. In the game $(G,\psi)$, the players are allowed to share arbitrarily many copies of $\psi$. The ...

This corrigendum corrects errors found by Jérémie Dusart in the proof of correctness of the algorithms in [D. G. Corneil, B. Dalton, and M. Habib, SIAM J. Comput., 42 (2013), pp. 792--807]; there are no changes in the algorithms themselves.

We present a spectral analysis of a continuous scaling algorithm for matrix scaling and operator scaling. The main result is that if the input matrix or operator has a spectral gap, then a natural gradient flow has linear convergence. This implies that a ...

The matroid parity (or matroid matching) problem, introduced as a common generalization of matching and matroid intersection problems, is so general that it requires an exponential number of oracle calls. Nevertheless, Lovász [Acta Sci. Math., 42 (1980), ...

Let $G=(V,E)$ be a weighted graph or multigraph, with $f$ or $b$ a function assigning a nonnegative integer to each vertex. An $f$-factor is a subgraph whose degree function is $f$; a perfect $b$-matching is a $b$-factor in the graph formed from $G$ by ...

We show that variants of spectral sparsification routines can preserve the total spanning tree counts of graphs. By Kirchhoff's matrix-tree theorem, this is equivalent to preserving the determinant of a graph Laplacian minor or, equivalently, of any symmetric ...

We show that if the nearly linear time solvers for Laplacian matrices and their generalizations can be extended to solve just slightly larger families of linear systems, then they can be used to quickly solve all systems of linear equations over the reals. ...

One of the simplest problems on directed graphs is that of identifying the set of vertices reachable from a designated source vertex. This problem can be solved easily sequentially by performing a graph search, but efficient parallel algorithms have eluded ...

Let $S$ be a set of $n$ points on a polyhedral terrain $\mathcal{T}$ in $\mathbb{R}^3$, and let $\varepsilon>0$ be a fixed constant. We prove that $S$ admits a $(2+\varepsilon)$-spanner with $O(n\log n)$ edges with respect to the geodesic distance. This is ...

Clustering is a classic topic in optimization with $k$-means being one of the most fundamental such problems. In the absence of any restrictions on the input, the best-known algorithm for $k$-means in Euclidean space with a provable guarantee is a simple ...

We study the computational complexity of short sentences in Presburger arithmetic (Short-PA). Here by “short” we mean sentences with a bounded number of variables, quantifiers, inequalities, and Boolean operations; the input consists only of the integer ...

The most well-known and ubiquitous clustering problem encountered in nearly every branch of science is undoubtedly $k$-means: given a set of data points and a parameter $k$, select $k$ centers and partition the data points into $k$ clusters around these ...

We study high-dimensional distribution learning in an agnostic setting where an adversary is allowed to arbitrarily corrupt an $\varepsilon$-fraction of the samples. Such questions have a rich history spanning statistics, machine learning, and theoretical ...

We develop a framework for the rigorous analysis of focused stochastic local search algorithms. These algorithms search a state space by repeatedly selecting some constraint that is violated in the current state and moving to a random nearby state that ...

We describe a fully dynamic linear-space data structure for point location in connected planar subdivisions, or more generally vertical ray shooting among nonintersecting line segments, that supports queries in $O(\log n(\log\log n)^2)$ time and updates ...

Harrow, Hassidim, and Lloyd [Phys. Rev. Lett., 103 (2009), 150502] showed that for a suitably specified $N \times N$ matrix $A$ and an $N$-dimensional vector $\vec{b}$, there is a quantum algorithm that outputs a quantum state proportional to the ...

Given an $n$-vertex graph and two straight-line planar drawings of the graph that have the same faces and the same outer face, we show that there is a morph (i.e., a continuous transformation) between the two drawings that preserves straight-line ...

In this paper we study variants of the widely used spectral clustering that partitions a graph into $k$ clusters by (1) embedding the vertices of a graph into a low-dimensional space using the bottom eigenvectors of the Laplacian matrix and (2) grouping ...

We study the role that parallelism plays in time complexity of variants of Winfree's abstract Tile Assembly Model (aTAM), a model of molecular algorithmic self-assembly. In the “hierarchical” aTAM, two assemblies, both consisting of multiple tiles, are ...