A Scaling Algorithm for Weighted f-Factors in General Graphs
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We study the maximum weight perfect f-factor problem on any general simple graph G=(V,E,w) with positive integral edge weights w, and n=|V|, m=|E|. When we have a function f:V→N_+ on vertices, a perfect f-factor is a generalized matching so that every vertex u is matched to f(u) different edges. The previous best algorithms on this problem have running time O(m f(V)) [Gabow 2018] or Õ(W(f(V))^2.373)) [Gabow and Sankowski 2013], where W is the maximum edge weight, and f(V)=∑_u∈ Vf(u). In this paper, we present a scaling algorithm for this problem with running time Õ(mn^2/3log W). Previously this bound is only known for bipartite graphs [Gabow and Tarjan 1989]. The running time of our algorithm is independent of f(V), and consequently it first breaks the Ω(mn) barrier for large f(V) even for the unweighted f-factor problem in general graphs.
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