Approximation algorithms and hardness results for the clique packing problem

F. Chataigner, G. Manić, Y. Wakabayashi, R. Yuster

Research output: Contribution to journalArticlepeer-review

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 = {K2}. In this paper we provide new approximation algorithms and hardness results for the Kr-packing problem where Kr = {K2, K3, ..., Kr}. We show that already for r = 3 the Kr-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 languageEnglish
Pages (from-to)1396-1406
Number of pages11
JournalDiscrete Applied Mathematics
Volume157
Issue number7
DOIs
StatePublished - 6 Apr 2009

Keywords

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

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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