SIAM Journal on Discrete Mathematics
Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
Packing triangles in bounded degree graphs
Information Processing Letters
Approximating k-Set Cover and Complementary Graph Coloring
Proceedings of the 5th International IPCO Conference on Integer Programming and Combinatorial Optimization
On linear and semidefinite programming relaxations for hypergraph matching
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Complexity results for linear XSAT-Problems
SAT'10 Proceedings of the 13th international conference on Theory and Applications of Satisfiability Testing
XSAT and NAE-SAT of linear CNF classes
Discrete Applied Mathematics
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For a fixed family F of graphs, an F-packing in 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}. In this paper we provide new approximation algorithms and hardness results for the K"r-packing problem where K"r={K"2,K"3,...,K"r}. We show that already for r=3 the K"r-packing problem is APX-complete, and, in fact, we show that 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 simple3/2-approximation, achieving a known ratio that follows from a more involved algorithm of Halldorsson. For r=4, we obtain a (3/2+@e)-approximation, and for r=5 we obtain a (25/14+@e)-approximation.