## Abstract

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 Halldórsson. For r = 4, we obtain a (3 / 2 + ε{lunate})-approximation, and for r = 5 we obtain a (25 / 14 + ε{lunate})-approximation.

Original language | English |
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Pages (from-to) | 1396-1406 |

Number of pages | 11 |

Journal | Discrete Applied Mathematics |

Volume | 157 |

Issue number | 7 |

DOIs | |

State | Published - 6 Apr 2009 |

## Keywords

- Approximation algorithms
- Clique
- Packing
- Triangle
- apx-hardness

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics