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

Authors Nick Fischer , Marvin Künnemann, Mirza Redžić , Julian Stieß

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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Graph algorithms analysis
Keywords
  • fine-grained complexity theory
  • clique detections in hypergraphs
  • constraint satisfaction
  • parameterized algorithms

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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 ≤ 1, we obtain a linear-time solvable problem, if d = 2, the time complexity is essentially equivalent to k-clique detection, and if d ≥ 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.

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Nick Fischer, Marvin Künnemann, Mirza Redžić, and Julian Stieß. The Role of Regularity in (Hyper-)Clique Detection and Implications for Optimizing Boolean CSPs. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 78:1-78:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.78

Author Details

Nick Fischer
  • INSAIT, Sofia University "St. Kliment Ohridski", Bulgaria
Marvin Künnemann
  • Karlsruhe Institute of Technology, Germany
Mirza Redžić
  • Karlsruhe Institute of Technology, Germany
Julian Stieß
  • Karlsruhe Institute of Technology, Germany

Funding

Research supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 462679611.
  • Fischer, Nick: Partially funded by the Ministry of Education and Science of Bulgaria’s support for INSAIT, Sofia University "St. Kliment Ohridski" as part of the Bulgarian National Roadmap for Research Infrastructure. Parts of this work were done while the author was at Weizmann Institute of Science.

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