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

Authors Shuai Shao , Stanislav Živný

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Shuai Shao
  • School of Computer Science and Technology, University of Chinese Academy of Sciences, Beijing, China
Stanislav Živný
  • Department of Computer Science, University of Oxford, UK

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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)


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.

Subject Classification

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


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