Finding a Maximum Clique in a Set of Proper Circular Arcs in Time O(n) with Applications

Abstract

A maximum clique is sought in a set of n proper circular arcs (PCAS). By means of several passes, each O(n) in time and space, a PCAS is transformed initially into a set of circle chords and finally into a set of intervals. This interval model inherits a special property from the PCAS which ensures the discovery of a maximum overlap clique in time O(n). The one-to-one arc/interval correspondence guarantees the identification of the maximum clique in the PCAS in O(n) time and space. The present paper gives new, simpler proofs for the lemmas first outlined by us in Ref. [9], extending the methods outlined in that paper so that the time bound is improved from O(n log n) to O(n). The method depends only on certain interconnections between constructions related to the computation of longest increasing subsequences. Independently, Hell, Huang and Bhattacharya5 recently discovered a completely different approach that also achieves the same complexity, and can moreover be applied to the weighted case and to the coloring problem on proper circular arcs. The previous best result, due to Apostolico and Hambrusch2 applies to general circular arc models and has time complexity O(n2 log log n) and space complexity O(n). As applications of the method, we show that maximum weight clique of a set of weighted proper circular arcs can be found in time O(n2) and space O(n). The previous best result was O(n2 log log n) for dense general circular arc graphs.13 We also show that, for n chords with randomly placed endpoints (1) the average cardinality of a maximum clique is cn1/2 ± o(n1/2), where 21/2< c < e21/2, and (2) a maximum clique may be found in average time O(n3/2) and space θ(n). The previous best average time complexity, derived from Ref. [1], was O(n3/2 log n).