k best cuts for circular-arc graphs

Abstract

Given a set of n weighted circular-arcs on a unit circle, and an integer k, the k Best Cuts for Circular-Arcs problem, abbreviated as k-BCCA problem, is to find a placement of k points, called cuts, on the circle such that the total weight of the arcs that contain at least one cut is maximized. We first solve a simpler version, the k Best Cuts for Intervals (k-BCI) problem in O(kn + n log n) time. We then define the k Restricted Best Cuts for Intervals (k-RBCI) problem, and solve it in the same complexity of k-BCI algorithm. These two algorithms are then used as subroutines to solve the k-BCCA problem in O(kn + n log n) or O(I(k)+ n log n) time, where I(k) is the time complexity of k-BCI problem. We also show that if k > 1, the k Maximum Cliques Cover problem for circular-arc graphs can be solved in O(I(k) + n log n) time.