2026-08-25T14:43:39Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:27251 2026-08-25T13:18:19Z ddc:004 open_access

Tight Bounds for Clique-Packing Parameterized by Clique-Width Bojikian, Narek Kratsch, Stefan Parameterized complexity triangle packing clique packing clique-width In the d-Clique Packing problem, given a graph G and an integer t, we need to decide whether G contains a set of t pairwise vertex-disjoint cliques of size d each. This generalizes Triangle Packing and it is NP-complete for all d ≥ 3. For each such d, we show how to solve the problem in n^𝒪(k^{d-1}) time where k is the clique-width of the graph (with a k-expression of G given in the input). We complement this by showing that, assuming the Exponential-Time Hypothesis (ETH), there is no algorithm that solves the problem in n^o(k^{d-1}) time for any fixed d ≥ 3, already for the special case of seeking a partition into cliques of size d. Our proof also entails W[1]-hardness of d-Clique Packing (and d-Clique Partitioning) parameterized by clique-width for each d ≥ 3. Our work continues a series of results on ETH-tight bounds for fundamental graph problems started by Fomin et al. (SICOMP 2010+2014) who obtained tight bounds for Max-Cut and Edge Dominating Set. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Narek Bojikian and Stefan Kratsch 2026 Is Part Of LIPIcs, Volume 388, 34th Annual European Symposium on Algorithms (ESA 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ESA.2026.115 urn:nbn:de:0030-drops-272519 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.115 eng https://creativecommons.org/licenses/by/4.0/legalcode