Given a set ofn nonnegativeweighted circular arcs on a unit circle, and an integerk, thek Best Cust for Circular-Arcs problem, abbreviated as thek-BCCA problem, is to find a placement ofk points, calledcuts, 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, thek Best Cuts for Intervals (k-BCI) problem, inO(kn+n logn) time andO(kn) space using dynamic programming. The algorithm is then extended to solve a variation, called thek-restricted BCI problem, and the space complexity of thek-BCI problem can be improved toO(n). Based on these results, we then show that thek-BCCA problem can be solved inO(I(k,n)+nlogn) time, whereI(k, n) is the time complexity of thek-BCI problem. As a by-product, thek Maximum Cliques Cover problem (k>1) for the circular-arc graphs can be solved inO(I(k,n)+nlogn) time.
Key WordsCircular-arc graph Interval graph Facility location Competitive location Maximum clique cover
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