Tight bounds for clique-packing parameterized by clique-width

2026-06-30

AI Summary

This study addresses the d-Clique Packing problem (for d ≥ 3), which asks whether a given graph contains t pairwise vertex-disjoint cliques of size d. Focusing on the parameterized complexity with respect to the graph’s clique-width k, the work establishes the first tight lower bound under the Exponential Time Hypothesis (ETH). It proves that the problem is W[1]-hard when parameterized by clique-width and presents an algorithm running in time n^{O(k^{d−1})}. Moreover, it shows that no algorithm can solve the problem in time n^{o(k^{d−1})} unless ETH fails. These results also apply to the special case of d-Clique Partition, where the vertex set must be partitioned into disjoint d-cliques.

Abstract

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 $d\geq 3$. 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 $d\geq 3$, 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 Partition) parameterized by clique-width for each $d\geq 3$. 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.

Problem

• Clique Packing
• Clique-width
• Parameterized Complexity
• Exponential-Time Hypothesis
• NP-complete

Innovation

• clique-packing
• clique-width
• ETH-tight bounds
• parameterized complexity
• W[1]-hardness