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

Authors: Qilong Feng, Guanlan Tan, Senmin Zhu, Bin Fu, and Jianxin Wang

Published in: LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)


Abstract
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 König-Egerváry 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 König-Egerváry 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.

Cite as

Qilong Feng, Guanlan Tan, Senmin Zhu, Bin Fu, and Jianxin Wang. New Algorithms for Edge Induced König-Egerváry Subgraph Based on Gallai-Edmonds Decomposition. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 31:1-31:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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@InProceedings{feng_et_al:LIPIcs.ISAAC.2018.31,
  author =	{Feng, Qilong and Tan, Guanlan and Zhu, Senmin and Fu, Bin and Wang, Jianxin},
  title =	{{New Algorithms for Edge Induced K\"{o}nig-Egerv\'{a}ry Subgraph Based on Gallai-Edmonds Decomposition}},
  booktitle =	{29th International Symposium on Algorithms and Computation (ISAAC 2018)},
  pages =	{31:1--31:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-094-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{123},
  editor =	{Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.31},
  URN =		{urn:nbn:de:0030-drops-99790},
  doi =		{10.4230/LIPIcs.ISAAC.2018.31},
  annote =	{Keywords: K\"{o}nig-Egerv\'{a}ry graph, Gallai-Edmonds decomposition}
}
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