2 research outputs found

    Maximum Bipartite Subgraph of Geometric Intersection Graphs

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    We study the Maximum Bipartite Subgraph (MBS) problem, which is defined as follows. Given a set SS of nn geometric objects in the plane, we want to compute a maximum-size subset Sβ€²βŠ†SS'\subseteq S such that the intersection graph of the objects in Sβ€²S' is bipartite. We first give a simple O(n)O(n)-time algorithm that solves the MBS problem on a set of nn intervals. We also give an O(n2)O(n^2)-time algorithm that computes a near-optimal solution for the problem on circular-arc graphs. We show that the MBS problem is NP-hard on geometric graphs for which the maximum independent set is NP-hard (hence, it is NP-hard even on unit squares and unit disks). On the other hand, we give a PTAS for the problem on unit squares and unit disks. Moreover, we show fast approximation algorithms with small-constant factors for the problem on unit squares, unit disks and unit-height rectangles. Finally, we study a closely related geometric problem, called Maximum Triangle-free Subgraph (TFS), where the objective is the same as that of MBS except the intersection graph induced by the set Sβ€²S' needs to be triangle-free only (instead of being bipartite).Comment: 32 pages, 7 figure

    The Generalized Independent and Dominating Set Problems on Unit Disk Graphs

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    In this article, we study a generalized version of the maximum independent set and minimum dominating set problems, namely, the maximum dd-distance independent set problem and the minimum dd-distance dominating set problem on unit disk graphs for a positive integer d>0d>0. We first show that the maximum dd-distance independent set problem and the minimum dd-distance dominating set problem belongs to NP-hard class. Next, we propose a simple polynomial-time constant-factor approximation algorithms and PTAS for both the problems
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