New results on maximum induced matchings in bipartite graphs and beyond

The maximum induced matching problem is known to be APX-hard in the class of bipartite graphs. Moreover, the problem is also intractable in this class from a parameterized point of view, i.e. it is W[1]-hard. In this paper, we reveal several classes of bipartite (and more general) graphs for which the problem admits fixed-parameter tractable algorithms. We also study the computational complexity of the problem for regular bipartite graphs and prove that the problem remains APX-hard even under this restriction. On the other hand, we show that for hypercubes (a proper subclass of regular bipartite graphs) the problem admits a simple solution.

[1]  Gerd Finke,et al.  Approximability results for the maximum and minimum maximal induced matching problems , 2008, Discret. Optim..

[2]  Michele Zito.  Induced Matchings in Regular Graphs and Trees , 1999, WG.

[3]  William Duckworth,et al.  A PTAS for the sparsest 2-spanner of 4-connected planar triangulations , 2003, J. Discrete Algorithms.

[4]  Andreas Brandstädt,et al.  The induced matching and chain subgraph cover problems for convex bipartite graphs , 2007, Theor. Comput. Sci..

[5]  Hannes Moser,et al.  The parameterized complexity of the induced matching problem , 2009, Discret. Appl. Math..

[6]  Paul C. Kainen,et al.  A lower bound for crossing numbers of graphs with applications to Kn, Kp,q, and Q(d) , 1972 .

[7]  Kathie Cameron,et al.  Induced Matchings in Intersection Graphs , 2000, Electron. Notes Discret. Math..

[8]  William Duckworth,et al.  Maximum Induced Matchings of Random Cubic Graphs , 2000, COCOON.

[9]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[10]  Hilda Assiyatun,et al.  Maximum Induced Matchings of Random Regular Graphs , 2003, IJCCGGT.

[11]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[12]  Jörg Flum,et al.  Parameterized Complexity Theory , 2006, Texts in Theoretical Computer Science. An EATCS Series.

[13]  Miroslav Chlebík,et al.  Approximation hardness of dominating set problems in bounded degree graphs , 2008, Inf. Comput..

[14]  Irena Rusu.  Maximum weight edge-constrained matchings , 2008, Discret. Appl. Math..

[15]  Vijay V. Vazirani,et al.  NP-Completeness of Some Generalizations of the Maximum Matching Problem , 1982, Inf. Process. Lett..

[16]  David Manlove,et al.  On the approximability of the maximum induced matching problem , 2005, J. Discrete Algorithms.

[17]  Martin Charles Golumbic,et al.  Irredundancy in Circular Arc Graphs , 1993, Discret. Appl. Math..

[18]  William Duckworth,et al.  Maximum induced matchings of random cubic graphs , 2002 .

[19]  Moshe Lewenstein,et al.  New results on induced matchings , 2000, Discret. Appl. Math..

[20]  Matthias Mnich,et al.  Induced Matchings in Subcubic Planar Graphs , 2012, SIAM J. Discret. Math..

[21]  Vassilis Giakoumakis,et al.  Bipartite Graphs Totally Decomposable by Canonical Decomposition , 1999, Int. J. Found. Comput. Sci..

[22]  Vadim V. Lozin.  On maximum induced matchings in bipartite graphs , 2002, Inf. Process. Lett..

[23]  Hannes Moser,et al.  Parameterized complexity of finding regular induced subgraphs , 2009, J. Discrete Algorithms.

[24]  Kathie Cameron,et al.  Finding a maximum induced matching in weakly chordal graphs , 2003, Discret. Math..

[25]  Angelika Steger,et al.  On induced matchings , 1993, Discret. Math..

[26]  Jörg Flum,et al.  Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series) , 2006 .

[27]  Jou-Ming Chang,et al.  Induced matchings in asteroidal triple-free graphs , 2003, Discret. Appl. Math..

[28]  Bruce L. Golden,et al.  Optimisation , 1982, IEEE Trans. Syst. Man Cybern..

[29]  Vassilis Giakoumakis,et al.  Linear Time Recognition and Optimizations for Weak-Bisplit Graphs, Bi-Cographs and Bipartite P6-Free Graphs , 2003, Int. J. Found. Comput. Sci..

[30]  Moshe Lewenstein,et al.  Tighter Approximations for Maximum Induced Matchings in Regular Graphs , 2005, WAOA.

[31]  Giorgio Gambosi,et al.  Complexity and Approximation , 1999, Springer Berlin Heidelberg.

[32]  Giorgio Gambosi,et al.  Complexity and approximation: combinatorial optimization problems and their approximability properties , 1999 .

[33]  Andreas Brandstädt,et al.  Maximum Induced Matchings for Chordal Graphs in Linear Time , 2008, Algorithmica.

[34]  Udi Rotics,et al.  Finding Maximum Induced Matchings in Subclasses of Claw-Free and P5-Free Graphs, and in Graphs with Matching and Induced Matching of Equal Maximum Size , 2003, Algorithmica.

[35]  F. Bruce Shepherd,et al.  Bipartite Domination and Simultaneous Matroid Covers , 2003, SIAM J. Discret. Math..

[36]  Zsolt Tuza,et al.  Induced matchings in bipartite graphs , 1989, Discret. Math..

[37]  R. Forcade,et al.  Smallest maximal matchings in the graph of the d-dimensional cube , 1973 .

[38]  F. Harary,et al.  A survey of the theory of hypercube graphs , 1988 .