PTAS for H-free node deletion problems in disk graphs

Abstract For a set H of graphs, a graph G is H -free if G does not contain any subgraph isomorphic to some graph in H . In this paper, we study the minimum H -free node deletion problem (Min H FND) and the maximum H -free node set problem (Max H FNS), which include a lot of extensively-studied problems such as the minimum k -path vertex cover problem, the dissociation number problem, and the minimum degree bounded node deletion problem. For a large class of H , PTASs are given for Min H FND and Max H FNS on disk graphs whose heterogeneity is bounded by a constant, where the heterogeneity of a disk graph is the ratio of the maximum radius to the minimum radius of disks.

[1]  Reuven Bar-Yehuda,et al.  A Local-Ratio Theorem for Approximating the Weighted Vertex Cover Problem , 1983, WG.

[2]  Bang Ye Wu,et al.  A Measure and Conquer Approach for the Parameterized Bounded Degree-One Vertex Deletion , 2015, COCOON.

[3]  Zhao Zhang,et al.  PTAS for minimum k-path vertex cover in ball graph , 2017, Inf. Process. Lett..

[4]  Zhao Zhang,et al.  A PTAS for the minimum weight connected vertex cover P3 problem on unit disk graphs , 2015, Theor. Comput. Sci..

[5]  Nabil H. Mustafa,et al.  Improved Results on Geometric Hitting Set Problems , 2010, Discret. Comput. Geom..

[6]  Marián Novotný,et al.  Design and Analysis of a Generalized Canvas Protocol , 2010, WISTP.

[7]  Ingo Schiermeyer,et al.  On computing the minimum 3-path vertex cover and dissociation number of graphs , 2011, Theor. Comput. Sci..

[8]  Rolf Niedermeier,et al.  On Bounded-Degree Vertex Deletion parameterized by treewidth , 2012, Discret. Appl. Math..

[9]  Weili Wu,et al.  Minimum vertex cover in ball graphs through local search , 2013, Journal of Global Optimization.

[10]  Zhao Zhang,et al.  Approximation algorithm for the minimum weight connected k-subgraph cover problem , 2014, Theor. Comput. Sci..

[11]  Mihalis Yannakakis,et al.  Node-Deletion Problems on Bipartite Graphs , 1981, SIAM J. Comput..

[12]  Jianhua Tu,et al.  The vertex cover P3P3 problem in cubic graphs , 2013, Inf. Process. Lett..

[13]  Fahad Panolan,et al.  Faster Parameterized Algorithms for Deletion to Split Graphs , 2012, Algorithmica.

[14]  Xiaohui Huang,et al.  Approximation algorithms for minimum (weight) connected k-path vertex cover , 2016, Discret. Appl. Math..

[15]  Vangelis Th. Paschos,et al.  Completeness in standard and differential approximation classes: Poly-(D)APX- and (D)PTAS-completeness , 2005, Theor. Comput. Sci..

[16]  Marko Jakovac,et al.  On the k-path vertex cover of some graph products , 2013, Discret. Math..

[17]  Ding-Zhu Du,et al.  Connected Dominating Sets in Wireless Networks with Different Transmission Ranges , 2007, IEEE Transactions on Mobile Computing.

[18]  Rastislav Krivos-Bellus,et al.  On the weighted k-path vertex cover problem , 2014, Discret. Appl. Math..

[19]  Peter Rossmanith,et al.  Fixed-parameter algorithms for vertex cover P3 , 2016, Discret. Optim..

[20]  Wenli Zhou,et al.  A factor 2 approximation algorithm for the vertex cover P3 problem , 2011, Inf. Process. Lett..

[21]  Bostjan Bresar,et al.  On the vertex kk-path cover , 2013, Discret. Appl. Math..

[22]  Mingyu Xiao,et al.  Faster Computation of the Maximum Dissociation Set and Minimum 3-Path Vertex Cover in Graphs , 2015, FAW.

[23]  Jianhua Tu,et al.  A 2-approximation algorithm for the vertex cover P4 problem in cubic graphs , 2014, Int. J. Comput. Math..

[24]  Wei Wang,et al.  PTAS for the minimum k-path connected vertex cover problem in unit disk graphs , 2013, J. Glob. Optim..

[25]  Saket Saurabh,et al.  Approximation algorithms for node deletion problems on bipartite graphs with finite forbidden subgraph characterization , 2014, Theor. Comput. Sci..

[26]  Jianhua Tu,et al.  A fixed-parameter algorithm for the vertex cover P3 problem , 2015, Inf. Process. Lett..

[27]  Wenli Zhou,et al.  A primal-dual approximation algorithm for the vertex cover P3 problem , 2011, Theor. Comput. Sci..

[28]  Amnon Barak,et al.  A new approach for approximating node deletion problems , 2003, Inf. Process. Lett..

[29]  Bostjan Bresar,et al.  Minimum k-path vertex cover , 2010, Discret. Appl. Math..

[30]  Zhao Zhang,et al.  A PTAS for minimum weighted connected vertex cover P3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_3$$\end{documen , 2015, Journal of Combinatorial Optimization.

[31]  Marko Jakovac,et al.  The k-path vertex cover of rooted product graphs , 2015, Discret. Appl. Math..

[32]  Narsingh Deo,et al.  Node-Deletion NP-Complete Problems , 1979, SIAM J. Comput..

[33]  Toshihiro Fujito,et al.  A unified approximation algorithm problems ” , 1998 .

[34]  John M. Lewis,et al.  The Node-Deletion Problem for Hereditary Properties is NP-Complete , 1980, J. Comput. Syst. Sci..