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Notes on Equitable Partitions into Matching Forests in Mixed Graphs and b-branchings in Digraphs

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Combinatorial Optimization (ISCO 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12176))

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Abstract

An equitable partition into branchings in a digraph is a partition of the arc set into branchings such that the sizes of any two branchings differ at most by one. For a digraph whose arc set can be partitioned into k branchings, there always exists an equitable partition into k branchings. In this paper, we present two extensions of equitable partitions into branchings in digraphs: those into matching forests in mixed graphs; and into b-branchings in digraphs. For matching forests, Király and Yokoi (2018) considered a tricriteria equitability based on the sizes of the matching forest, and the matching and branching therein. In contrast to this, we introduce a single-criterion equitability based on the number of the covered vertices. For b-branchings, we define an equitability based on the size of the b-branching and the indegrees of all vertices. For both matching forests and b-branchings, we prove that equitable partitions always exist.

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Notes

  1. 1.

    We believe that this notation causes no confusion on the direction of the arcs, since we never refer to the set of the tails of the arcs in this paper.

References

  1. Baum, S., Trotter Jr., L.E.: Integer rounding for polymatroid and branching optimization problems. SIAM J. Algebraic Discrete Methods 2, 416–425 (1981)

    Article  MathSciNet  Google Scholar 

  2. Bock, F.: An algorithm to construct a minimum directed spanning tree in a directed network. In: Developments in Operations Research, pp. 29–44. Gordon and Breach (1971)

    Google Scholar 

  3. Bouchet, A.: Greedy algorithm and symmetric matroids. Math. Program. 38, 147–159 (1987)

    Article  MathSciNet  Google Scholar 

  4. Chandrasekaran, R., Kabadi, S.N.: Pseudomatroids. Discrete Math. 71, 205–217 (1988)

    Article  MathSciNet  Google Scholar 

  5. Chu, Y.J., Liu, T.H.: On the shortest arborescence of a directed graph. Sci. Sinica 14, 1396–1400 (1965)

    MathSciNet  MATH  Google Scholar 

  6. Davies, J., McDiarmid, C.: Disjoint common transversals and exchange structures. J. London Math. Soc. 14, 55–62 (1976)

    Article  MathSciNet  Google Scholar 

  7. de Werra, D.: On some combinatorial problems arising in scheduling. Can. Oper. Res. J. 8, 165–175 (1970)

    MathSciNet  MATH  Google Scholar 

  8. de Werra, D.: Decomposition of bipartite multigraphs into matchings. Zeitschrift für Oper. Res. 16, 85–90 (1972)

    MathSciNet  MATH  Google Scholar 

  9. Dress, A.W.M., Havel, T.: Some combinatorial properties of discriminants in metric vector spaces. Adv. Math. 62, 285–312 (1986)

    Article  MathSciNet  Google Scholar 

  10. Dulmage, A.L., Mendelsohn, N.S.: Some graphical properties of matrices with non-negative entries. Aequationes Math. 2, 150–162 (1969)

    Article  MathSciNet  Google Scholar 

  11. Edmonds, J.: Edge-disjoint branchings. In: Rustin, R. (ed.) Combinatorial Algorithms, pp. 91–96. Algorithmics Press (1973)

    Google Scholar 

  12. Edmonds, J.: Optimum branchings. J. Res. Natl. Bureau Stand. Sect. B 71, 233–240 (1967)

    Article  MathSciNet  Google Scholar 

  13. Erdős, P.: Problem 9. In: Fiedler, M. (ed.) Theory of Graphs and its Applications, p. 159. Czech Academy of Sciences (1964)

    Google Scholar 

  14. Folkman, J., Fulkerson, D.R.: Edge colorings in bipartite graphs. In: Bose, R.C., Dowling, T.A. (eds.) Combinatorial Mathematics and Its Applications (Proceedings Conference Chapel Hill, North Carolina, 1967), pp. 561–577. The University of North Carolina Press, Chapel Hill (1969)

    Google Scholar 

  15. Fujishige, S., Takazawa, K., Yokoi, Y.: A note on a nearly uniform partition into common independent sets of two matroids. J. Oper. Res. Soc. Japan (to appear)

    Google Scholar 

  16. Fulkerson, D.R.: Packing rooted directed cuts in a weighted directed graph. Math. Program. 6(1), 1–13 (1974)

    Article  MathSciNet  Google Scholar 

  17. Giles, R.: Optimum matching forests I: special weights. Math. Program. 22, 1–11 (1982)

    Article  MathSciNet  Google Scholar 

  18. Giles, R.: Optimum matching forests II: general weights. Math. Program. 22, 12–38 (1982)

    Article  MathSciNet  Google Scholar 

  19. Giles, R.: Optimum matching forests III: facets of matching forest polyhedra. Math. Program. 22, 39–51 (1982)

    Article  MathSciNet  Google Scholar 

  20. Hajnal, A., Szemerédi, E.: Proof of a conjecture of P. Erdős. In: Erdős, P., Rényi, A., Sós, V. (eds.) Combinatorial Theory and its Applications, II (Proceedings Colloquium on Combinatorial Theory and its Applications, Balatonfüred, Hungary, 1969), North-Holland, Amsterdam, pp. 601–623 (1970)

    Google Scholar 

  21. Kakimura, N., Kamiyama, N., Takazawa, K.: The \(b\)-branching problem in digraphs. Discrete Appl. Math. (to appear)

    Google Scholar 

  22. Keijsper, J.: A Vizing-type theorem for matching forests. Discrete Math. 260, 211–216 (2003)

    Article  MathSciNet  Google Scholar 

  23. Király, T., Yokoi, Y.: Equitable partitions into matchings and coverings in mixed graphs. CoRR abs/1811.07856 (2018)

    Google Scholar 

  24. McDiarmid, C.J.H.: The solution of a timetabling problem. J. Inst. Math. Appl. 9, 23–34 (1972)

    Article  MathSciNet  Google Scholar 

  25. McDiarmid, C.: Integral decomposition in polyhedra. Math. Program. 25(2), 183–198 (1983)

    Article  MathSciNet  Google Scholar 

  26. Schrijver, A.: Total dual integrality of matching forest constraints. Combinatorica 20, 575–588 (2000)

    Article  MathSciNet  Google Scholar 

  27. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Heidelberg (2003)

    MATH  Google Scholar 

  28. Takazawa, K.: Optimal matching forests and valuated delta-matroids. SIAM J. Discrete Math. 28, 445–467 (2014)

    Article  MathSciNet  Google Scholar 

  29. Takazawa, K.: The \(b\)-bibranching problem: TDI system, packing, and discrete convexity. CoRR abs/1802.03235 (2018)

    Google Scholar 

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Acknowledgements

This work is partially supported by JSPS KAKENHI Grant Numbers JP16K16012, JP20K11699, Japan.

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Authors and Affiliations

  1. Department of Industrial and Systems Engineering, Faculty of Science and Engineering, Hosei University, 3-7-2, Kajino-cho, Koganei-shi, Tokyo, 184-8584, Japan

    Kenjiro Takazawa

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  1. Kenjiro Takazawa
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Correspondence to Kenjiro Takazawa .

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Editors and Affiliations

  1. CNRS and Université Clermont Auvergne, Aubière, France

    Mourad Baïou

  2. Département d'informatique et de recherche opérationnelle & CIRRELT, University of Montreal, Montréal, QC, Canada

    Bernard Gendron

  3. School of Operations Research and Information Engineering, Cornell University, Ithaca, NY, USA

    Oktay Günlük

  4. Laboratoire LAMSADE, Université Paris-Dauphine, Paris, France

    A. Ridha Mahjoub

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Takazawa, K. (2020). Notes on Equitable Partitions into Matching Forests in Mixed Graphs and b-branchings in Digraphs. In: Baïou, M., Gendron, B., Günlük, O., Mahjoub, A.R. (eds) Combinatorial Optimization. ISCO 2020. Lecture Notes in Computer Science(), vol 12176. Springer, Cham. https://doi.org/10.1007/978-3-030-53262-8_18

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  • DOI: https://doi.org/10.1007/978-3-030-53262-8_18

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  • Print ISBN: 978-3-030-53261-1

  • Online ISBN: 978-3-030-53262-8

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