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 = {K₂}. We provide new approximation algorithms and hardness results for the K_r-packing problem where K_r = {K₂, K₃, ..., 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{lunate} and 25 / 14 + ε{lunate}, respectively.