On the k-coloring of intervals Original Research Article

The problem of coloring a set of n intervals (from the real line) with a set of k colors is studied. In such a coloring, two intersecting intervals must receive distinct colors. Our main result is an O(k + n) algorithm for k-coloring a maximum cardinality subset of the intervals, assuming that the endpoints of the intervals are presorted. Previous methods are linear only in n, and assume that k is a fixed constant. In addition to the main result, we provide an O(kS(n)) algorithm for k-coloring a set of weighted intervals of maximum total weight. Here, S(n) is the running time of any algorithm for finding shortest paths in graphs with O(n) edges. The best previous algorithm for this problem required time O(nS(n)). Since in most applications, k is substantially smaller than n, the saving is significant.