General factors are a generalization of matchings. Given a graph G with a set π(v) of feasible degrees, called a degree constraint, for each vertex v of G, the general factor problem is to find a (spanning) subgraph F of G such that deg_F(v) ∈ π(v) for every v of G. When all degree constraints are symmetric Δ-matroids, the problem is solvable in polynomial time. The weighted general factor problem is to find a general factor of the maximum total weight in an edge-weighted graph. Strongly polynomial-time algorithms are only known for weighted general factor problems that are reducible to the weighted matching problem by gadget constructions. In this paper, we present a strongly polynomial-time algorithm for a type of weighted general factor problems with real-valued edge weights that is provably not reducible to the weighted matching problem by gadget constructions. As an application, we obtain a strongly polynomial-time algorithm for the terminal backup problem by reducing it to the weighted general factor problem.
@InProceedings{shao_et_al:LIPIcs.ISAAC.2023.57, author = {Shao, Shuai and \v{Z}ivn\'{y}, Stanislav}, title = {{A Strongly Polynomial-Time Algorithm for Weighted General Factors with Three Feasible Degrees}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {57:1--57:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.57}, URN = {urn:nbn:de:0030-drops-193597}, doi = {10.4230/LIPIcs.ISAAC.2023.57}, annote = {Keywords: matchings, factors, edge constraint satisfaction problems, terminal backup problem, delta matroids} }
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