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Approximate Counting CSP Seen from the Other Side

Published:13 May 2020Publication History
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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.

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      • Published in

        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 12, Issue 2
        June 2020
        138 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/3382781
        Issue’s Table of Contents

        Copyright © 2020 ACM

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        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 13 May 2020
        • Online AM: 7 May 2020
        • Accepted: 1 January 2020
        • Revised: 1 December 2019
        • Received: 1 July 2019
        Published in toct Volume 12, Issue 2

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