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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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Author Details

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

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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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

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.

Subject Classification

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

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References

  1. Jin Akiyama and Mikio Kano. Factors and factorizations of graphs: Proof techniques in factor theory, volume 2031. Springer, 2011. Google Scholar
  2. Elliot Anshelevich and Adriana Karagiozova. Terminal backup, 3d matching, and covering cubic graphs. SIAM Journal on Computing, 40(3):678-708, 2011. Google Scholar
  3. Richard P. Anstee. A polynomial algorithm for b-matchings: an alternative approach. Information Processing Letters, 24(3):153-157, 1987. Google Scholar
  4. André Bouchet. Greedy algorithm and symmetric matroids. Mathematical Programming, 38(2):147-159, 1987. URL: https://doi.org/10.1007/BF02604639.
  5. André Bouchet. Matchings and Δ-matroids. Discrete Applied Mathematics, 24(1-3):55-62, 1989. Google Scholar
  6. David A Cohen, Martin C Cooper, Peter G Jeavons, and Andrei A Krokhin. The complexity of soft constraint satisfaction. Artificial Intelligence, 170(11):983-1016, 2006. Google Scholar
  7. Gérard Cornuéjols. General factors of graphs. Journal of Combinatorial Theory, Series B, 45(2):185-198, 1988. URL: https://doi.org/10.1016/0095-8956(88)90068-8.
  8. William H. Cunningham and Alfred B. Marsh. A primal algorithm for optimum matching. In Polyhedral Combinatorics, pages 50-72. Springer, 1978. Google Scholar
  9. Víctor Dalmau and Daniel K. Ford. Generalized satisfability with limited occurrences per variable: A study through delta-matroid parity. In Proceedings of the 28th International Symposium on Mathematical Foundations of Computer Science (MFCS'03), volume 2747, pages 358-367. Springer, 2003. URL: https://doi.org/10.1007/978-3-540-45138-9_30.
  10. Ran Duan and Seth Pettie. Linear-time approximation for maximum weight matching. Journal of the ACM, 61(1):1-23, 2014. URL: https://doi.org/10.1145/2529989.
  11. Szymon Dudycz and Katarzyna Paluch. Optimal general matchings. Lecture Notes in Computer Science, 11159 LNCS:176-189, 2018. URL: https://doi.org/10.1007/978-3-030-00256-5_15.
  12. Szymon Dudycz and Katarzyna Paluch. Optimal general matchings. arXiv, version 3, 2021. URL: https://arxiv.org/abs/1706.07418v3.
  13. Zdeněk Dvořák and Martin Kupec. On Planar Boolean CSP. In Proceedings of the 42nd International Colloquium on Automata, Languages and Programming (ICALP'15), volume 9134, pages 432-443. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-47672-7_35.
  14. Jack Edmonds. Maximum matching and a polyhedron with 0, 1-vertices. Journal of research of the National Bureau of Standards B, 69(125-130):55-56, 1965. Google Scholar
  15. Jack Edmonds. Paths, trees, and flowers. Canadian Journal of mathematics, 17:449-467, 1965. URL: https://doi.org/10.4153/CJM-1965-045-4.
  16. Jack Edmonds and Ellis L Johnson. Matching: A well-solved class of integer linear programs. In Combinatorial Structures and Their Applications, pages 89-92. Gordon & Breach, New York, 1970. Google Scholar
  17. Tomás Feder. Fanout limitations on constraint systems. Theoretical Computer Science, 255(1-2):281-293, 2001. URL: https://doi.org/10.1016/S0304-3975(99)00288-1.
  18. Tomás Feder and Daniel K. Ford. Classification of bipartite Boolean constraint satisfaction through Delta-matroid intersection. SIAM J. Discrete Math., 20(2):372-394, 2006. URL: https://doi.org/10.1137/S0895480104445009.
  19. Tomás Feder and Moshe Y. Vardi. The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory. SIAM Journal on Computing, 28(1):57-104, 1998. URL: https://doi.org/10.1137/S0097539794266766.
  20. Harold N. Gabow. Implementation of algorithms for maximum matching on nonbipartite graphs. PhD thesis, Stanford University, 1974. Google Scholar
  21. Harold N. Gabow. An efficient reduction technique for degree-constrained subgraph and bidirected network flow problems. In Proceedings of the fifteenth annual ACM symposium on Theory of computing, pages 448-456, 1983. Google Scholar
  22. Harold N. Gabow. A scaling algorithm for weighted matching on general graphs. In Proceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science (FOCS'85), pages 90-100, 1985. Google Scholar
  23. Harold N. Gabow. Data structures for weighted matching and nearest common ancestors with linking. In Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms, pages 434-443, 1990. Google Scholar
  24. Harold N. Gabow, Zvi Galil, and Thomas H. Spencer. Efficient implementation of graph algorithms using contraction. Journal of the ACM (JACM), 36(3):540-572, 1989. Google Scholar
  25. Harold N. Gabow and Piotr Sankowski. Algebraic algorithms for b-matching, shortest undirected paths, and f-factors. In Proceedings of the 54th IEEE Annual Symposium on Foundations of Computer Science (FOCS'13), pages 137-146, 2013. Google Scholar
  26. Harold N. Gabow and Robert E. Tarjan. Faster scaling algorithms for general graph matching problems. Journal of the ACM (JACM), 38(4):815-853, 1991. Google Scholar
  27. Zvi Galil, Silvio Micali, and Harold N. Gabow. An O(EV$1log V) algorithm for finding a maximal weighted matching in general graphs. SIAM Journal on Computing, 15(1):120-130, 1986. Google Scholar
  28. James F. Geelen, Satoru Iwata, and Kazuo Murota. The linear delta-matroid parity problem. Journal of Combinatorial Theory, Series B, 88(2):377-398, 2003. URL: https://doi.org/10.1016/S0095-8956(03)00039-X.
  29. Chien-Chung Huang and Telikepalli Kavitha. Efficient algorithms for maximum weight matchings in general graphs with small edge weights. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1400-1412. SIAM, 2012. Google Scholar
  30. Gabriel Istrate. Looking for a version of Schaefer’s dichotomy theorem when each variable occurs at most twice. Technical report, University of Rochester, 1997. UR CSD/TR652. Google Scholar
  31. Aleksandr V. Karzanov. Efficient implementations of Edmonds’ algorithms for finding matchings with maximum cardinality and maximum weight. Studies in Discrete Optimization, pages 306-327, 1976. Google Scholar
  32. Alexandr Kazda, Vladimir Kolmogorov, and Michal Rolínek. Even delta-matroids and the complexity of planar Boolean CSPs. ACM Transactions on Algorithms (TALG), 15(2):1-33, 2018. Google Scholar
  33. Yusuke Kobayashi. Optimal general factor problem and jump system intersection. In International Conference on Integer Programming and Combinatorial Optimization, pages 291-305. Springer, 2023. Google Scholar
  34. Bernhard Korte and Jens Vygen. Combinatorial optimization: Theory and Algorithms, volume 21. Springer, 2018. Google Scholar
  35. Eugene L. Lawler. Combinatorial optimization: networks and matroids. Holt, Reinhart and Winston, New York., 1976. Google Scholar
  36. László Lovász. The factorization of graphs. In Combinatorial Structures and Their Applications, pages 243-246. Gordon & Breach, New York, 1970. Google Scholar
  37. László Lovász. The factorization of graphs. II. Acta Mathematica Academiae Scientiarum Hungarica, 23(1-2):223-246, 1972. URL: https://doi.org/10.1007/BF01889919.
  38. László Lovász and Michael D. Plummer. Matching theory, volume 367. American Mathematical Soc., 2009. Google Scholar
  39. Alfred B. Marsh. Matching algorithms. PhD thesis, The Johns Hopkins University, 1979. Google Scholar
  40. Michael D. Plummer. Graph factors and factorization: 1985-2003: a survey. Discrete Mathematics, 307(7-8):791-821, 2007. Google Scholar
  41. William R. Pulleyblank. Faces of Matching Polyhedra. PhD thesis, University of Waterloo, 1973. Google Scholar
  42. Thomas J. Schaefer. The complexity of satisfiability problems. In Proceedings of the tenth annual ACM symposium on Theory of computing, pages 216-226, 1978. Google Scholar
  43. Alexander Schrijver. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer, 2003. Google Scholar
  44. Shuai Shao and Stanislav Živný. A Strongly Polynomial-Time Algorithm for Weighted General Factors with Three Feasible Degrees. arXiv, 2023. URL: https://arxiv.org/abs/2301.11761.
  45. Jácint Szabó. Good characterizations for some degree constrained subgraphs. Journal of Combinatorial Theory, Series B, 99(2):436-446, 2009. URL: https://doi.org/10.1016/J.JCTB.2008.08.009.
  46. William Thomas Tutte. The factorization of locally finite graphs. Canadian Journal of Mathematics, 2:44-49, 1950. Google Scholar
  47. William Thomas Tutte. The factors of graphs. Canadian Journal of Mathematics, 4:314-328, 1952. Google Scholar
  48. William Thomas Tutte. A short proof of the factor theorem for finite graphs. Canadian Journal of mathematics, 6:347-352, 1954. Google Scholar
  49. Dahai Xu, Elliot Anshelevich, and Mung Chiang. On survivable access network design: Complexity and algorithms. In IEEE INFOCOM 2008-The 27th Conference on Computer Communications, pages 186-190. IEEE, 2008. Google Scholar
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