New clique and independent set algorithms for circle graphs

Given the interval model of an n-venex, e-edge circle graph G. it is shown how to find a clique of G of maximum size 1 Cresp.. maximum weight) in o (nlogn+min[e,nllog(2nll)]) (resp., O(nlogn+min[n 2,eloglogn])) time. The best previous algoritfuns required. respectively, Ben2) and O(n 2+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 !.he 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 0 (n 3). • This research was supponed in part by the French and Ilalian MinisDies of Education, by the British Research Council Grant SERC-E76797. by NSF Grant CCR-89-00305, by NIH Librlll)' of Medicine Grant ROl LM05I18, by AFOSR GranL 89NM682., and by NATO GranL eRG 900293. u 'This research wo.s supponed in part by the Office of Naval Research under Contracts NOOOI4-84·K· 0502 and NOOOI4-86-K-0689. the Air Force Office of Scientific Research under Grant AFOSR-90-0l07. the National Science Foundation under Grant DCR-8451393. and the National Library of Medicine under Grant ROl-LM05118. U. This research was supponed in pan by the Office of Naval Research. under Contracts NOOOI4-S4·K· 0502 and NOOO14-86-K-0689, and by the National Science Foundalion under Gr:m.t MIP.g7·15652.