Approximate Counting CSP Seen from the Other Side

Andrei A Bulatov,Stanislav Živný

doi:10.1145/3389390

Abstract

In this article, we study the complexity of counting Constraint Satisfaction Problems (CSPs) of the form #CSP( C , -), in which the goal is, given a relational structure A from a class C of structures and an arbitrary structure B , to find the number of homomorphisms from A to B . Flum and Grohe showed that #CSP( C , -) is solvable in polynomial time if C has bounded treewidth [FOCS’02]. Building on the work of Grohe [JACM’07] on decision CSPs, Dalmau and Jonsson then showed that if C is a recursively enumerable class of relational structures of bounded arity, then, assuming FPT≠ #W[1], there are no other cases of #CSP( C , -) solvable exactly in polynomial time (or even fixed-parameter time) [TCS’04]. We show that, assuming FPT ≠ W[1] (under randomised parameterised reductions) and for C satisfying certain general conditions, #CSP( C ,-) is not solvable even approximately for C of unbounded treewidth; that is, there is no fixed parameter tractable (and thus also not fully polynomial) randomised approximation scheme for #CSP( C , -). In particular, our condition generalises the case when C is closed undertaking minors.

Highlights

The Constraint Satisfaction Problem (CSP) asks to decide the existence of a homomorphism between two given relational structures
Flum and Grohe showed that #CSP(C, −) is solvable in polynomial time if C has bounded treewidth [22]
We show that for C such that a certain class of graphs is a subset of G(C), #CSP(C, −) cannot be solved even approximately for C of unbounded treewidth, assuming FPT = W[1]

Summary

Introduction

The Constraint Satisfaction Problem (CSP) asks to decide the existence of a homomorphism between two given relational structures (or to find the number of such homomorphisms). Since the general CSP is NP-complete (#P-complete in the counting case) and because one needs to model specific computational problems, various restricted versions of the CSP have been considered. Let C and D be two classes of relational structures. The constraint satisfaction problem (CSP) parametrised by C and D is the following computational problem, denoted by CSP(C, D): given A ∈ C and B ∈ D, is there a homomorphism from A to B? CSPs in which both input structures are restricted have not received much attention (with a notable exception of matrix partitions [19, 20] and assorted graph problems on restricted classes of graphs).

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Conclusion