Abstract
Let F be a family of arcs in a circle, where each arc is associated with a weight. The maximum weight clique problem is to find a subset of pairwise overlapping arcs in F such that their total weight is maximum. This problem was originally solved by Hsu in O( m· n) time. Recently, an O( n 2 log log n) algorithm was discovered by Apostolico and Hambrusch for the unweighted case, whereas the weighted case was posed as an open problem. We extend their approach to solve the weighted case in O( n log n + m log log n) time. Our algorithm is based on a characterization of maximum weight independent sets in bipartite graphs, which is not related to bipartite matching.
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