Approximating the Maximum Independent Set and Minimum Vertex Coloring on Box Graphs

A box graph is the intersection graph of a finite set of orthogonal rectangles in the plane. The problem of whether or not the maximum independent set problem (MIS for short) for box graphs can be approximated within a substantially sub-logarithmic factor in polynomial time has been open for several years. We show that for box graphs on nvertices which have an independent set of size i¾?(n/logO(1)n) the maximum independent set problem can be approximated within O(logn/ loglogn) in polynomial time. Furthermore, we show that the chromatic number of a box graph on nvertices is within an O(logn) factor from the size of its maximum clique and provide an O(logn) approximation algorithm for minimum vertex coloring of such a box graph. More generally, we can show that the chromatic number of the intersection graph of nd-dimensional orthogonal rectangles is within an O(logdi¾? 1n) factor from the size of its maximum clique and obtain an O(logdi¾? 1n) approximation algorithm for minimum vertex coloring of such an intersection graph.

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