Given the interval model of an n-vertex, e-edge circle graph G, it is shown how to find a clique of G of maximum size l (respectively, maximum weight) in O(nlogn+min[e,nllog(2n/l)]) (respectively, O(nlogn+min[n2,eloglogn])) time. The best previous algorithms required, respectively, ¦](n2) and O(n2+eloglogn) time. An O(nlogn+dn) time and space algorithm that finds an independent set of maximum weight for the interval model of G is also presented. Here d is the maximum number of intervals crossing any position of the line in the interval model of G. The best previous solution for this problem took time O(n3).
