k best cuts for circular-arc graphsArticle Received: 05 March 1995 Revised: 21 December 1995 DOI :
10.1007/BF02526033

Cite this article as: Tsai, K.H. & Lee, D.T. Algorithmica (1997) 18: 198. doi:10.1007/BF02526033
Abstract 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) +n logn ) 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) +n logn ) time.

Key Words Circular-arc graph Interval graph Facility location Competitive location Maximum clique cover This work was supported in part by the National Science Foundation under Grants CCR-8901815, CCR-9309743, and INT-9207212, and by the Office of Naval Research under Grant No. N00014-93-1-0272.

Communicated by C. K. Wong.

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Authors and Affiliations 1. Institute of Information Science Academia Sinica Taipei Taiwan 2. Department of Electrical and Computer Engineering Northwestern University Evanston USA