A Note on Approximating Weighted Independence on Intersection Graphs of Paths on a Grid
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A graph G is called B_k-VPG, for some constant k≥ 0, if it has a string representation on an axis-parallel grid such that each vertex is a path with at most k bends and two vertices are adjacent in G if and only if the corresponding paths intersect each other. The part of a path that is between two consecutive bends is called a segment of the path. In this paper, we study the Maximum-Weighted Independent Set problem on B_k-VPG graphs. The problem is known to be NP-complete on B_1-VPG graphs, even when the two segments of every path have unit length [12], and O( n)-approximation algorithms are known on B_k-VPG graphs, for k≤ 2 [3, 14]. In this paper, we give a (ck+c+1)-approximation algorithm for the problem on B_k-VPG graphs for any k≥ 0, where c>0 is the length of the longest segment among all segments of paths in the graph. Notice that c is not required to be a constant; for instance, when c∈ O( n), we get an O( n)-approximation or we get an O(1)-approximation when c is a constant. To our knowledge, this is the first o( n)-approximation algorithm for a non-trivial subclass of B_k-VPG graphs.
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