## Abstract

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{lunate} and 25 / 14 + ε{lunate}, respectively.

Original language | English |
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Pages (from-to) | 397-401 |

Number of pages | 5 |

Journal | Electronic Notes in Discrete Mathematics |

Volume | 29 |

Issue number | SPEC. ISS. |

DOIs | |

State | Published - 15 Aug 2007 |

## Keywords

- APX-hardness
- Approximation algorithms
- clique
- packing
- triangle

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics