Approximating MIS over equilateral B_1-VPG graphs
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We present an approximation algorithm for the maximum independent set (MIS) problem over the class of equilateral B_1-VPG graphs. These are intersection graphs of L-shaped planar objects 90^o) with both arms of each object being equal. We obtain a 36(log 2d)-approximate algorithm running in O(n(log n)^2) time for this problem, where d is the ratio d_max/d_min and d_max and d_min denote respectively the maximum and minimum length of any arm in the input equilateral L-representation of the graph. In particular, we obtain O(1)-factor approximation of MIS for B_1-VPG -graphs for which the ratio d is bounded by a constant. generalized to an O(n(log n)^2) time and a 36(log 2d_x)(log 2d_y)-approximate MIS algorithm over arbitrary B_1-VPG graphs. Here, d_x and d_y denote respectively the analogues of d when restricted to only horizontal and vertical arms of members of the input. This is an improvement over the previously best n^ϵ-approximate algorithm <cit.> (for some fixed ϵ>0), unless the ratio d is exponentially large in n. In particular, O(1)-approximation of MIS is achieved for graphs with max{d_x,d_y}=O(1).
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