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 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.
Blogger's Review: This paper makes significant progress in the complexity analysis of the Clique Packing problem, particularly in the parameterization by clique-width. By introducing the ETH assumption, the authors not only provide time complexity bounds for the algorithm but also reveal the inherent difficulty of the problem, offering strong theoretical support for research in related fields.