AMS/MBA 546 (Fall, 2010)Estie ArkinNetwork FlowsHomework Set # 9 (last)Due Thursday, December 9, 2010.1). Stable marriages with an unequal number of men and women: LetWbe the set of women andMbethe set of men, and assume that|W|<|M|. A matching is unstable if there is a manmand womanwsuchthat:•mandware not partners in the matching;•mis either unmarried, or preferswto his wife in the matching;•wis either unmarried, or prefersmto her husband in the matching.Prove that a stable marriage always exists.2). Given an undirected graphG= (N, E), recall thatχ(G) is the (node) colouring number ofG(the fewestcolours needed to colour all nodes ofG), andα(G) is the size of the largest independent (stable) set (subsetof nodes no two of which are connected by an edge). The following parts are independet of each other.(a). Show thatχ(G)·α(G)≥nwherenis the number of nodes inG.(b). LetGbe a tree. Give a polynomial time algorithm for computingα(G).(c). LetGbe a planar graph. Give a polynomial time approximation algorithm that returns an independent
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