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The Complexity of Counting in Sparse, Regular, and Planar Graphs

Author:

Salil P. VadhanAuthors Info & Claims

SIAM Journal on Computing, Volume 31, Issue 2

Pages 398 - 427

https://doi.org/10.1137/S0097539797321602

Published: 01 February 2002 Publication History

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Abstract

We show that a number of graph-theoretic counting problems remain ${\cal NP}$-hard, indeed $\#{\cal P}$-complete, in very restricted classes of graphs. In particular, we prove that the problems of counting matchings, vertex covers, independent sets, and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree. We obtain corollaries about counting cliques in restricted classes of graphs and counting satisfying assignments to restricted classes of monotone 2-CNF formulae. To achieve these results, a new interpolation-based reduction technique which preserves properties such as constant degree is introduced.

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Bonnet ÉChakraborty DDuron J(2024)Cutting Barnette graphs perfectly is hardTheoretical Computer Science10.1016/j.tcs.2024.1147011010:COnline publication date: 27-Sep-2024
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Matsuoka TOhsaka N(2024)Computational complexity of normalizing constants for the product of determinantal point processesTheoretical Computer Science10.1016/j.tcs.2024.114513997:COnline publication date: 27-May-2024
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Fan ACai J(2023)Dichotomy result on 3-regular bipartite non-negative functionsTheoretical Computer Science10.1016/j.tcs.2023.113745949:COnline publication date: 9-Mar-2023
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Index Terms

The Complexity of Counting in Sparse, Regular, and Planar Graphs
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Enumeration
    2. Graph theory
2. Theory of computation
  1. Computational complexity and cryptography
    1. Problems, reductions and completeness
  2. Randomness, geometry and discrete structures

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Published In

SIAM Journal on Computing Volume 31, Issue 2

2001

334 pages

ISSN:0097-5397

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Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 February 2002

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