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A Strongly Polynomial-Time Algorithm for Weighted General Factors with Three Feasible Degrees

Authors Shuai Shao , Stanislav Živný

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  • Theory of computation → Design and analysis of algorithms
Keywords
  • matchings
  • factors
  • edge constraint satisfaction problems
  • terminal backup problem
  • delta matroids

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Abstract

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.

Cite As Get BibTex

Shuai Shao and Stanislav Živný. A Strongly Polynomial-Time Algorithm for Weighted General Factors with Three Feasible Degrees. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 57:1-57:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ISAAC.2023.57

Author Details

Shuai Shao
  • School of Computer Science and Technology & Hefei National Laboratory, University of Science and Technology of China, Hefei, China
Stanislav Živný
  • Department of Computer Science, University of Oxford, UK

Funding

The research leading to these results has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 714532). This research was also funded by UKRI EP/X024431/1. All data is provided in full in the results section of this paper. Part of the work was done while the first author was a postdoctoral research associate at the University of Oxford.

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