Approximating the maximum independent set and minimum vertex coloring on box graphs

Abstract

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 n vertices which have an independent set of size Ω(n/ logO(1) n) the maximum independent set problem can be approximated within O(log n/ log log n) in polynomial time. Furthermore, we show that the chromatic number of a box graph on n vertices is within an O(log n) 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 n ddimensional orthogonal rectangles is within an O(logd_1 n) factor from the size of its maximum clique and obtain an O(logd-1 n) approximation algorithm for minimum vertex coloring of such an intersection graph. © Springer-Verlag Berlin Heidelberg 2007.