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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, Jianxin Wang

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Author Details

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

Funding

This work is supported by the National Natural Science Foundation of China under Grants (61420106009, 61872450, 61828205, 61672536).

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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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