Approximation algorithms and hardness results for the clique packing problem

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

Research output: Contribution to journalArticlepeer-review


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 Kr-packing problem where Kr = {K2, K3, ..., Kr}. We prove that already for r = 3 the Kr-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 languageEnglish
Pages (from-to)397-401
Number of pages5
JournalElectronic Notes in Discrete Mathematics
Issue numberSPEC. ISS.
StatePublished - 15 Aug 2007


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

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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