New Algorithms for Edge Induced König-Egerváry Subgraph Based on Gallai-Edmonds Decomposition

König-Egerváry graphs form an important graph class which has been studied extensively in graph theory. Much attention has also been paid on König-Egerváry subgraphs and KönigEgerváry graph modification problems. In this paper, we focus on one König-Egerváry subgraph problem, called the Maximum Edge Induced König Subgraph problem. By exploiting the classical Gallai-Edmonds decomposition, we establish connections between minimum vertex cover, Gallai-Edmonds decomposition structure, maximum matching, maximum bisection, and KönigEgerváry subgraph structure. We obtain a new structural property of König-Egerváry subgraph: every graph G = (V,E) has an edge induced König-Egerváry subgraph with at least 2|E|/3 edges. Based on the new structural property proposed, an approximation algorithm with ratio 10/7 for the Maximum Edge Induced König Subgraph problem is presented, improving the current best ratio of 5/3. To the best of our knowledge, this paper is the first one establishing the connection between Gallai-Edmonds decomposition and König-Egerváry graphs. Using 2|E|/3 as a lower bound, we define the Edge Induced König Subgraph above lower bound problem, and give a kernel of at most 30k edges for the problem. 2012 ACM Subject Classification Mathematics of computing → Graph algorithms, Mathematics of computing → Approximation algorithms

[1]  Michael Lampis.  A kernel of order 2 k-c log k for vertex cover , 2011, Inf. Process. Lett..

[2]  Jianer Chen,et al.  Constrained minimum vertex cover in bipartite graphs: complexity and parameterized algorithms , 2003, J. Comput. Syst. Sci..

[3]  Vadim E. Levit,et al.  On maximum matchings in König-Egerváry graphs , 2013, Discret. Appl. Math..

[4]  Michal Pilipczuk,et al.  On Multiway Cut Parameterized above Lower Bounds , 2011, IPEC.

[5]  William R. Pulleyblank,et al.  König-Egerváry graphs, 2-bicritical graphs and fractional matchings , 1989, Discret. Appl. Math..

[6]  Shankar M. Venkatesan,et al.  Approximation and Intractability Results for the Maximum Cut Problem and its Variants , 1991, IEEE Trans. Computers.

[7]  Hiroshi Nagamochi,et al.  A 2-approximation algorithm for the minimum weight edge dominating set problem , 2002, Discret. Appl. Math..

[8]  Saket Saurabh,et al.  König Deletion Sets and Vertex Covers above the Matching Size , 2008, ISAAC.

[9]  Vadim E. Levit,et al.  Two more characterizations of König-Egerváry graphs , 2017, Discret. Appl. Math..

[10]  Claudio L. Lucchesi,et al.  Ear Decompositions of Matching Covered Graphs , 1999, Comb..

[11]  Saket Saurabh,et al.  The Complexity of Finding Subgraphs Whose Matching Number Equals the Vertex Cover Number , 2007, ISAAC.

[12]  Bo Ji,et al.  Throughput characterization of node-based scheduling in multihop wireless networks: a novel application of the Gallai-Edmonds structure theorem , 2016, MobiHoc.

[13]  Saket Saurabh,et al.  On approximability of optimization problems related to Red/Blue-split graphs , 2017, Theor. Comput. Sci..

[14]  Saket Saurabh,et al.  The Complexity of König Subgraph Problems and Above-Guarantee Vertex Cover , 2010, Algorithmica.

[15]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[16]  F Stersoul,et al.  A characterization of the graphs in which the transversal number equals the matching number , 1979, J. Comb. Theory, Ser. B.

[17]  Stefan Kratsch,et al.  A randomized polynomial kernelization for Vertex Cover with a smaller parameter , 2016, ESA.

[18]  Eric McDermid.  A 3/2-Approximation Algorithm for General Stable Marriage , 2009, ICALP.

[19]  J. Edmonds.  Paths, Trees, and Flowers , 1965, Canadian Journal of Mathematics.

[20]  Saket Saurabh,et al.  Faster Parameterized Algorithms Using Linear Programming , 2012, ACM Trans. Algorithms.

[21]  Saket Saurabh,et al.  LP can be a cure for Parameterized Problems , 2012, STACS.

[22]  Miklós Bartha,et al.  Molecular Switching by Turing Automata , 2011, NCMA.

[23]  Martin Milanic,et al.  On two extensions of equimatchable graphs , 2016, Discret. Optim..

[24]  Barry O'Sullivan,et al.  Almost 2-SAT is Fixed-Parameter Tractable , 2008, J. Comput. Syst. Sci..

[25]  Saket Saurabh,et al.  Paths, Flowers and Vertex Cover , 2011, ESA.

[26]  Jianxin Wang,et al.  A New Kernel for Parameterized Max-Bisection Above Tight Lower Bound , 2017, COCOON.

[27]  Matthias Mnich,et al.  Bisections above Tight Lower Bounds , 2012, WG.

[28]  Britta Peis,et al.  Subgraph characterization of red/blue-split graph and kőnig egerváry graphs , 2006, SODA '06.

[29]  Tayfun Sönmez,et al.  Altruistically Unbalanced Kidney Exchange , 2012, J. Econ. Theory.

[30]  Domingos M. Cardoso,et al.  Combinatorial and spectral properties of König-Egerváry graphs , 2017, Discret. Appl. Math..

[31]  Geevarghese Philip,et al.  Raising The Bar For Vertex Cover: Fixed-parameter Tractability Above A Higher Guarantee , 2015, SODA.

[32]  Ojas Parekh,et al.  Edge dominating and hypomatchable sets , 2002, SODA '02.

[33]  Francis Bloch,et al.  Efficient partnership formation in networks , 2019, Theoretical Economics.

[34]  Ge Xia,et al.  Improved Parameterized Upper Bounds for Vertex Cover , 2006, MFCS.

[35]  Vadim E. Levit,et al.  Critical Independent Sets and König–Egerváry Graphs , 2009, Graphs Comb..

[36]  Robert W. Deming,et al.  Independence numbers of graphs-an extension of the Koenig-Egervary theorem , 1979, Discret. Math..

[37]  Mitre Costa Dourado,et al.  Forbidden subgraphs and the König-Egerváry property , 2013, Discret. Appl. Math..