Packing $K_r$s in bounded degree graphs

TL;DR

This paper classifies the computational complexity of finding maximum sets of $r$-cliques in bounded degree graphs, extending known results for $r=3$ to all fixed $r \\geq 3$ and analyzing both vertex- and edge-disjoint cases.

Contribution

It provides a complete complexity classification for the maximum $r$-clique packing problem in graphs with bounded degree, generalizing prior specific cases.

Findings

Vertex-disjoint problem solvable in linear time if  < 3r/2 - 1.

Vertex-disjoint problem solvable in polynomial time if  < 5r/3 - 1.

APX-hard if      ceil - 1.

Abstract

We study the problem of finding a maximum-cardinality set of $r$-cliques in an undirected graph of fixed maximum degree $\Delta$, subject to the cliques in that set being either vertex-disjoint or edge-disjoint. It is known for $r=3$ that the vertex-disjoint (edge-disjoint) problem is solvable in linear time if $\Delta=3$ ($\Delta=4$) but APX-hard if $\Delta \geq 4$ ($\Delta \geq 5$). We generalise these results to an arbitrary but fixed $r \geq 3$, and provide a complete complexity classification for both the vertex- and edge-disjoint variants in graphs of maximum degree $\Delta$. Specifically, we show that the vertex-disjoint problem is solvable in linear time if $\Delta < 3r/2 - 1$, solvable in polynomial time if $\Delta < 5r/3 - 1$, and APX-hard if $\Delta \geq \lceil 5r/3 \rceil - 1$. We also show that if $r\geq 6$ then the above implications also hold for the edge-disjoint problem. If $r \leq 5$, then the edge-disjoint problem is solvable in linear time if $\Delta < 3r/2 - 1$, solvable in polynomial time if $\Delta \leq 2r - 2$, and APX-hard if $\Delta > 2r - 2$.