The Role of Regularity in (Hyper-)Clique Detection and Implications for Optimizing Boolean CSPs
Nick Fischer, Marvin Künnemann, Mirza Redžić, Julian Stieß
TL;DR
This work investigates whether strong regularity in $k$-partite $h$-uniform hypergraphs preserves the hardness of $k$-clique and $(k,h)$-Hyperclique detection, and shows a fine-grained equivalence between regular and general instances via a signed-product construction with template hypergraphs. It then uses this regularization to fully classify the fine-grained complexity of optimizing Boolean CSPs over weight-$k$ assignments, with a dichotomy governed by the maximum degree $d$ of the constraint polynomials: linear-time for $d\le 1$, $k$-clique-type bounds for $d=2$, and brute-force lower bounds under the $3$-uniform $k$-Hyperclique Hypothesis for $d\ge 3$. The results provide a robust framework for transferring hardness from general to regular instances and yield tight algorithmic boundaries for maxCSP$_k(\mathcal{F})$ across constraint families. The regularization technique also strengthens reductions and hints at broad applicability to future conditional lower bounds and design of new 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.