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

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



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Qilong Feng
  • School of Information Science and Engineering, Central South University, Changsha, P.R. China
Guanlan Tan
  • School of Information Science and Engineering, Central South University, Changsha, P.R. China
Senmin Zhu
  • School of Information Science and Engineering, Central South University, Changsha, P.R. China
Bin Fu
  • Department of Computer Science, University of Texas-Rio Grande Valley, USA
Jianxin Wang
  • School of Information Science and Engineering, Central South University, Changsha, P.R. China

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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)
https://doi.org/10.4230/LIPIcs.ISAAC.2018.31

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Approximation algorithms
Keywords
  • König-Egerváry graph
  • Gallai-Edmonds decomposition

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