AnO(m + n log n) Algorithm for the Maximum-Clique Problem in Circular-Arc Graphs

• Published in: Journal of algorithms Vol. 25; no. 2; pp. 336 - 358
• Main Authors: Bhattacharya, Binay K., Kaller, Damon
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-11-1997
• ISSN: 0196-6774; 1090-2678
• DOI: 10.1006/jagm.1997.0868

We present an algorithm to compute, inO(m+nlogn) time, a maximum clique in circular-arc graphs (withnvertices andmedges) provided a circular-arc model of the graph is given. If the circular-arc endpoints are given in sorted order, the time complexity isO(m). The algorithm operates on the geometric structure of the circular arcs, radially sweeping their endpoints; it uses a very simple data structure consisting of doubly linked lists. Previously, the best time bound for this problem wasO(mloglogn+nlogn), using an algorithm that solved an independent subproblem for each of thencircular arcs. By using the radial-sweep technique, we need not solve each of these subproblems independently; thus we eliminate the loglognfactor from the running time of earlier algorithms. For vertex-weighted circular-arc graphs, it is possible to use our approach to obtain anO(mloglogn+nlogn) algorithm for finding a maximum-weight clique—which matches the best known algorithm.