Inapproximability of Counting Hypergraph Colourings

Recent developments in approximate counting have made startling progress developing fast algorithmic methods for approximating the number of solutions to constraint satisfaction problems (CSPs) with large arities, using connections Lovász Local Lemma. Nevertheless, boundaries these CSPs non-Boolean domain are not well-understood. Our goal this article is fill gap and obtain strong inapproximability results by studying prototypical problem class CSPs, hypergraph colourings. More precisely, we focus on approximately q -colourings K -uniform hypergraphs bounded degree Δ. An efficient algorithm exists if \({{\Delta \lesssim \frac{q^{K/3-1}}{4^KK^2}}}\) [Jain et al. 25 ; He 23 ]. Somewhat surprisingly however, a hardness bound known even easier finding For problem, situation less clear there no evidence right constant controlling growth exponent terms . To end, first establish that general computational colouring simple/linear occurs at Δ ≳ Kq , almost matching from second main contribution far more refined goes well beyond which conjecture asymptotically tight (up factors). We show particular all ≥ 4 it NP -hard colourings when K/2 approach based considering an auxiliary weighted binary CSP model graphs, obtained “halving” -ary constraints. This allows us utilise reduction techniques available graph case, hinge upon understanding behaviour random regular bipartite graphs serve as gadgets reduction. The major challenge our setting analyse induced matrix norm interaction new captures most likely system. In contrast previous analyses literature, demonstrates both symmetry asymmetry, making analysis optimisation severely complicated demanding combination delicate perturbation arguments careful asymptotic estimates.