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Inapproximability of Counting Hypergraph Colourings

Published:01 February 2023Publication History
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Abstract

Recent developments in approximate counting have made startling progress in developing fast algorithmic methods for approximating the number of solutions to constraint satisfaction problems (CSPs) with large arities, using connections to the Lovász Local Lemma. Nevertheless, the boundaries of these methods for CSPs with non-Boolean domain are not well-understood. Our goal in this article is to fill in this gap and obtain strong inapproximability results by studying the prototypical problem in this class of CSPs, hypergraph colourings.

More precisely, we focus on the problem of approximately counting q-colourings on K-uniform hypergraphs with bounded degree Δ. An efficient algorithm exists if \({{\Delta \lesssim \frac{q^{K/3-1}}{4^KK^2}}}\) [Jain et al. 25; He et al. 23]. Somewhat surprisingly however, a hardness bound is not known even for the easier problem of finding colourings. For the counting problem, the situation is even less clear and there is no evidence of the right constant controlling the growth of the exponent in terms of K.

To this end, we first establish that for general q computational hardness for finding a colouring on simple/linear hypergraphs occurs at Δ ≳ KqK, almost matching the algorithm from the Lovász Local Lemma. Our second and main contribution is to obtain a far more refined bound for the counting problem that goes well beyond the hardness of finding a colouring and which we conjecture is asymptotically tight (up to constant factors). We show in particular that for all even q ≥ 4 it is NP-hard to approximate the number of colourings when Δ ≳ qK/2. Our approach is based on considering an auxiliary weighted binary CSP model on graphs, which is obtained by “halving” the K-ary hypergraph constraints. This allows us to utilise reduction techniques available for the graph case, which hinge upon understanding the behaviour of random regular bipartite graphs that serve as gadgets in the reduction. The major challenge in our setting is to analyse the induced matrix norm of the interaction matrix of the new CSP which captures the most likely solutions of the system. In contrast to previous analyses in the literature, the auxiliary CSP demonstrates both symmetry and asymmetry, making the analysis of the optimisation problem severely more complicated and demanding the combination of delicate perturbation arguments and careful asymptotic estimates.

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      • Published in

        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 14, Issue 3-4
        December 2022
        122 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/3582881
        Issue’s Table of Contents

        Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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        Publication History

        • Published: 1 February 2023
        • Online AM: 2 September 2022
        • Accepted: 20 August 2022
        • Revised: 17 August 2022
        • Received: 1 November 2021
        Published in toct Volume 14, Issue 3-4

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