MAXIMUM INDEPENDENT SET OF A PERMUTATION GRAPH IN K-TRACKS

A maximum independent set of a permutation graph is a maximum subset of noncrossing chords in a matching diagram PHI. (PHI consists of a set 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.