The Role of Regularity in (Hyper-)Clique Detection and Implications for Optimizing Boolean CSPs

TL;DR

This paper establishes a deep connection between regular hypergraph clique detection and the general case, and applies this to fully characterize the complexity of optimizing Boolean CSPs based on constraint degree.

Contribution

It proves a fine-grained equivalence between regular hyperclique detection and the general case, and characterizes Boolean CSP optimization complexity based on constraint degree.

Findings

Regular hyperclique detection is as hard as the general case.

Boolean CSP optimization complexity depends on maximum constraint degree.

The regularization technique is crucial for proving hardness results.

Abstract

Is detecting a $k$-clique in $k$-partite regular (hyper-)graphs as hard as in the general case? Intuition suggests yes, but proving this -- especially for hypergraphs -- poses notable challenges. Concretely, we consider a strong notion of regularity in $h$-uniform hypergraphs, where we essentially require that any subset of at most $h-1$ is incident to a uniform number of hyperedges. Such notions are studied intensively in the combinatorial block design literature. We show that any $f(k)n^{g(k)}$-time algorithm for detecting $k$-cliques in such graphs transfers to an $f'(k)n^{g(k)}$-time algorithm for the general case, establishing a fine-grained equivalence between the $h$-uniform hyperclique hypothesis and its natural regular analogue. Equipped with this regularization result, we then fully resolve the fine-grained complexity of optimizing Boolean constraint satisfaction problems over assignments with $k$ non-zeros. Our characterization depends on the maximum degree $d$ of a constraint function. Specifically, if $d\le 1$, we obtain a linear-time solvable problem, if $d=2$, the time complexity is essentially equivalent to $k$-clique detection, and if $d\ge 3$ the problem requires exhaustive-search time under the 3-uniform hyperclique hypothesis. To obtain our hardness results, the regularization result plays a crucial role, enabling a very convenient approach when applied carefully. We believe that our regularization result will find further applications in the future.