Approximation algorithms and hardness results for the clique packing problem

For a fixed family F of graphs, an F -packing of a graph G is a set of pairwise vertex-disjoint subgraphs of G, each isomorphic to an element of F . Finding an F -packing that maximizes the number of covered edges is a natural generalization of the maximum matching problem, which is just F = { K 2 } . We provide new approximation algorithms and hardness results for the K r -packing problem where K r = { K 2 , K 3 , … , K r } . We prove that already for r = 3 the K r -packing problem is APX-hard, and it remains so even for graphs with maximum degree 4. On the positive side, we give an approximation algorithm with approximation ratio at most 2 for every fixed r. For r = 3 , 4 , 5 we obtain better approximations. For r = 3 we obtain a simple 3/2 approximation, matching a known ratio that follows from a more complicated algorithm of Halldórsson. For r = 4 and r = 5 , we obtain the ratios 3 / 2 + ϵ and 25 / 14 + ϵ , respectively.