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 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.
König-Egerváry graph
Gallai-Edmonds decomposition
Mathematics of computing~Graph algorithms
Mathematics of computing~Approximation algorithms
31:1-31:12
Regular Paper
This work is supported by the National Natural Science Foundation of China under Grants (61420106009, 61872450, 61828205, 61672536).
Qilong
Feng
Qilong Feng
School of Information Science and Engineering, Central South University, Changsha, P.R. China
Guanlan
Tan
Guanlan Tan
School of Information Science and Engineering, Central South University, Changsha, P.R. China
Senmin
Zhu
Senmin Zhu
School of Information Science and Engineering, Central South University, Changsha, P.R. China
Bin
Fu
Bin Fu
Department of Computer Science, University of Texas-Rio Grande Valley, USA
Jianxin
Wang
Jianxin Wang
School of Information Science and Engineering, Central South University, Changsha, P.R. China
10.4230/LIPIcs.ISAAC.2018.31
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Qilong Feng, Guanlan Tan, Senmin Zhu, Bin Fu, and Jianxin Wang
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