Approximability results for the maximum and minimum maximal induced matching problems

  • Authors:

  • Yury Orlovich;Gerd Finke;Valery Gordon;Igor Zverovich

  • Affiliations:

  • Department of Combinatorial Models and Algorithms, Institute of Mathematics, National Academy of Sciences of Belarus, 11 Surganova Str., 220072 Minsk, Belarus;University Joseph Fourier, Laboratory G-SCOP, 46 Avenue Félix Viallet, 38031 Grenoble, France;United Institute of Informatics Problems, National Academy of Sciences of Belarus, 6 Surganova Str., 220012 Minsk, Belarus;RUTCOR-Rutgers Center for Operations Research, Rutgers, The State University of New Jersey, 640 Bartholomew Road, Piscataway, NJ 08854-8003, USA

  • Venue:
  • Discrete Optimization
  • Year:
  • 2008

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Abstract

An induced matchingM of a graph G is a set of pairwise non-adjacent edges such that their end-vertices induce a 1-regular subgraph. An induced matching M is maximal if no other induced matching contains M. The Minimum Maximal Induced Matching problem asks for a minimum maximal induced matching in a given graph. The problem is known to be NP-complete. Here we show that, if PNP, for any @e0, this problem cannot be approximated within a factor of n^1^-^@e in polynomial time, where n denotes the number of vertices in the input graph. The result holds even if the graph in question is restricted to being bipartite. For the related problem of finding an induced matching of maximum size (Maximum Induced Matching), it is shown that, if PNP, for any @e0, the problem cannot be approximated within a factor of n^1^/^2^-^@e in polynomial time. Moreover, we show that Maximum Induced Matching is NP-complete for planar line graphs of planar bipartite graphs.