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Complexity classification of counting graph homomorphisms modulo a prime number

Andrei A. Bulatov, Amirhossein Kazeminia

TL;DR

Techniques that widen the spectrum of reductions available for modular counting and apply to the general CSP rather than being limited to graph homomorphisms are developed.

Abstract

Counting graph homomorphisms and its generalizations such as the Counting Constraint Satisfaction Problem (CSP), its variations, and counting problems in general have been intensively studied since the pioneering work of Valiant. While the complexity of exact counting of graph homomorphisms (Dyer and Greenhill, 2000) and the counting CSP (Bulatov, 2013, and Dyer and Richerby, 2013) is well understood, counting modulo some natural number has attracted considerable interest as well. In their 2015 paper Faben and Jerrum suggested a conjecture stating that counting homomorphisms to a fixed graph H modulo a prime number is hard whenever it is hard to count exactly, unless H has automorphisms of certain kind. In this paper we confirm this conjecture. As a part of this investigation we develop techniques that widen the spectrum of reductions available for modular counting and apply to the general CSP rather than being limited to graph homomorphisms.

Complexity classification of counting graph homomorphisms modulo a prime number

TL;DR

Techniques that widen the spectrum of reductions available for modular counting and apply to the general CSP rather than being limited to graph homomorphisms are developed.

Abstract

Counting graph homomorphisms and its generalizations such as the Counting Constraint Satisfaction Problem (CSP), its variations, and counting problems in general have been intensively studied since the pioneering work of Valiant. While the complexity of exact counting of graph homomorphisms (Dyer and Greenhill, 2000) and the counting CSP (Bulatov, 2013, and Dyer and Richerby, 2013) is well understood, counting modulo some natural number has attracted considerable interest as well. In their 2015 paper Faben and Jerrum suggested a conjecture stating that counting homomorphisms to a fixed graph H modulo a prime number is hard whenever it is hard to count exactly, unless H has automorphisms of certain kind. In this paper we confirm this conjecture. As a part of this investigation we develop techniques that widen the spectrum of reductions available for modular counting and apply to the general CSP rather than being limited to graph homomorphisms.

Paper Structure

This paper contains 39 sections, 46 theorems, 105 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Homomorphisms and the Constraint Satisfaction Problem
  3. Counting CSP and graph homomorphisms
  4. Modular counting
  5. The existing results
  6. Outline of the proof
  7. $\#_{p}\mathsf{Hom}(H^{*p}) \leq_T \#_{p}\mathsf{Hom}(H)$
  8. $\#_p\mathsf{CSP}(\mathcal{H}^\mathsf{c})\leq_T \#_{p}\mathsf{Hom}(H^{*p})$
  9. $\#_p \mathsf{CSP}(\mathcal{H}^\mathsf{Z})\leq_T\#_p\mathsf{CSP}(\mathcal{H}^\mathsf{c})$
  10. Primitive positive definitions and polynomial time reductions
  11. Thick Z-Graphs and $\mathcal{H}^\mathsf{Z}$
  12. $\#_p\mathsf{BIS}(\alpha, \beta )\leq_T \#_p \mathsf{CSP}{(\mathcal{H}^\mathsf{Z}})$
  13. $\#_p\mathsf{CSP}(\mathcal{H}^{3\mathsf{SAT}})\le_T\#_p\mathsf{CSP}(\mathcal{H}^\mathsf{c})$
  14. $\#_p3\mathsf{SAT}\le_T\#_p\mathsf{CSP}(\mathcal{H}^{3\mathsf{SAT}})$
  15. $\#_p\mathsf{CSP}(\mathcal{H}^\dagger)\le_T\#_p\mathsf{CSP}(\mathcal{H}^\mathsf{c})$
  16. ...and 24 more sections

Key Result

Lemma 1.1

Let $H$ be a graph and $p$ a prime number. Up to an isomorphism there is a unique smallest (in terms of the number of vertices) graph $H^{*p}$ such that $H\rightarrow_p^* H^{*p}$, and for any graph $G$ it holds Moreover, $H^{*p}$ does not have automorphisms of order $p$.

Figures (5)

  • Figure 1: The chain of reductions
  • Figure 2: Thick Z-graph
  • Figure 3: Reduction of BIS
  • Figure 4: The structure of (a) thick star and (b) an independent 3-path.
  • Figure 5: The structure of $\mathcal{G}'$.

Theorems & Definitions (93)

  • Lemma 1.1: ref:CountingMod2Ini
  • Example 1.2
  • Conjecture 1.3: ref:CountingMod2Iniref:CountingModPToTrees_gbel_et_al_LIPIcs
  • Theorem 1.4
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 83 more