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 n vertices which have an independent set of size Ω(n/ log 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(log n) 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 d-dimensional orthogonal rectangles is within an O(log n) factor from the size of its maximum clique and obtain an O(log n) approximation algorithm for minimum vertex coloring of such an intersection graph.