MAXIMUM INDEPENDENT SET OF A PERMUTATION GRAPH IN K TRACKS

A maximum weighted independent set of a permutation graph is a maximum subset of noncrossing chords in a matching diagram (i.e., a set (Phi) of chords with end-points on two horizontal lines). The problem of finding, among all noncrossing subsets of Phi with density at most k, one with maximum size is considered, where the density of a subset is the maximum number of chords crossing a vertical line and k is a given parameter. A Theta(n log n) time and Theta(n) space algorithm, for solving the problem with n chords, is proposed. As an application, we solve the problem of finding, among all proper subsets with density at most k of an interval graph, one with maximum number of intervals.