*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.

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