We study the maximum weight perfect f-factor problem on any general simple graph G = (V,E,ω) 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 exactly f(u) different edges. The previous best results 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 ∈ V}f(u). In this paper, we present a scaling algorithm for this problem with running time Õ(mn^{2/3} log W). Previously this bound is only known for bipartite graphs [Gabow and Tarjan 1989]. The advantage is that the running time is independent of f(V), and consequently it breaks the Ω(mn) barrier for large f(V) even for the unweighted f-factor problem in general graphs.
@InProceedings{duan_et_al:LIPIcs.ICALP.2020.41, author = {Duan, Ran and He, Haoqing and Zhang, Tianyi}, title = {{A Scaling Algorithm for Weighted f-Factors in General Graphs}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {41:1--41:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.41}, URN = {urn:nbn:de:0030-drops-124487}, doi = {10.4230/LIPIcs.ICALP.2020.41}, annote = {Keywords: Scaling Algorithm, f-Factors, General Graphs} }
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