k Best Cuts for Circular-Arc graphs
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.
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