## Abstract

Given a set of*n* nonnegative*weighted* circular arcs on a unit circle, and an integer*k*, the*k* Best Cust for Circular-Arcs problem, abbreviated as the*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 and*O(kn)* space using dynamic programming. The algorithm is then extended to solve a variation, called the*k*-restricted BCI problem, and the space complexity of the*k*-BCI problem can be improved to*O(n)*. Based on these results, we then show that the*k*-BCCA problem can be solved in*O(I(k,n)*+*n*log*n*) time, where*I(k, n)* is the time complexity of the*k*-BCI problem. As a by-product, the*k* Maximum Cliques Cover problem (*k*>1) for the circular-arc graphs can be solved in*O(I(k,n)*+*n*log*n*) time.

### Key Words

Circular-arc graph Interval graph Facility location Competitive location Maximum clique cover## Preview

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