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Finding a Stable Allocation in Polymatroid Intersection

Author

Listed:
  • Satoru Iwata

    (Department of Mathematical Informatics, University of Tokyo, Tokyo 113-8656, Japan;)

  • Yu Yokoi

    (Principles of Informatics Research Division, National Institute of Informatics, Tokyo 101-8430, Japan)

Abstract

The stable matching (or stable marriage) model of Gale and Shapley [Gale D, Shapley LS (1962) College admissions and the stability of marriage. Amer. Math. Monthly 69(1):9–15.] has been generalized in various directions, such as matroid kernels by Fleiner [Fleiner T (2001) A matroid generalization of the stable matching polytope. Aardal K, Gerards AMH, eds. Proc. 8th Internat. Conf. Integer Programming Combin. Optim., Lecture Notes in Computer Science, vol. 2081 (Springer-Verlag, Berlin), 105–114.] and stable allocations in bipartite networks by Baïou and Balinski [Baïou M, Balinski M (2002) Erratum: The stable allocation (or ordinal transportation) problem. Math. Oper. Res. 27(4):662–680.]. Unifying these generalizations, we introduce the concept of stable allocations in polymatroid intersection. Our framework includes both integer and real variable versions. The integer variable version corresponds to a special case of the discrete concave function model of Eguchi et al. [Eguchi A, Fujishige S, Tamura A (2003) A generalized Gale-Shapley algorithm for a discrete-concave stable-marriage model. Ibaraki T, Katoh N, Ono H, eds. Proc. 14th Internat. Sympos. Algorithms Comput ., Lecture Notes in Computer Science, vol. 2906 (Springer-Verlag, Berlin), 495–504.], who established the existence of a stable allocation by showing that a simple extension of the deferred acceptance algorithm of Gale and Shapley finds a stable allocation in pseudopolynomial time. It has been open to develop a polynomial time algorithm even for our special case. In this paper, we present the first strongly polynomial algorithm for finding a stable allocation in polymatroid intersection. To achieve this, we utilize the augmenting path technique for polymatroid intersection. In each iteration, the algorithm searches for an augmenting path by simulating a chain of proposes and rejects in the deferred acceptance algorithm. The running time of our algorithm is O ( n 3 γ), where n and γ denote the cardinality of the ground set and the time for computing the saturation and exchange capacities, respectively. Moreover, we show that the output of our algorithm is optimal for one side, where this optimality is a generalization of the man optimality in the stable marriage model.

Suggested Citation

  • Satoru Iwata & Yu Yokoi, 2020. "Finding a Stable Allocation in Polymatroid Intersection," Mathematics of Operations Research, INFORMS, vol. 45(1), pages 63-85, February.
    • Handle: RePEc:inm:ormoor:v:45:y:2020:i:1:p:63-85
    DOI: 10.1287/moor.2018.0976
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    References listed on IDEAS

    as
    1. Tamás Fleiner, 2003. "A Fixed-Point Approach to Stable Matchings and Some Applications," Mathematics of Operations Research, INFORMS, vol. 28(1), pages 103-126, February.
    2. Alkan, Ahmet & Gale, David, 2003. "Stable schedule matching under revealed preference," Journal of Economic Theory, Elsevier, vol. 112(2), pages 289-306, October.
    3. Yu Yokoi, 2017. "A Generalized Polymatroid Approach to Stable Matchings with Lower Quotas," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 238-255, January.
    4. Éva Tardos & Craig A. Tovey & Michael A. Trick, 1986. "Layered Augmenting Path Algorithms," Mathematics of Operations Research, INFORMS, vol. 11(2), pages 362-370, May.
    5. Mourad Baïou & Michel Balinski, 2002. "The Stable Allocation (or Ordinal Transportation) Problem," Mathematics of Operations Research, INFORMS, vol. 27(3), pages 485-503, August.
    6. Mourad Baïou & Michel Balinski, 2002. "Erratum: The Stable Allocation (or Ordinal Transportation) Problem," Mathematics of Operations Research, INFORMS, vol. 27(4), pages 662-680, November.
    7. Satoru Fujishige & Akihisa Tamura, 2007. "A Two-Sided Discrete-Concave Market with Possibly Bounded Side Payments: An Approach by Discrete Convex Analysis," Mathematics of Operations Research, INFORMS, vol. 32(1), pages 136-155, February.
    8. Kazuo Murota & Yu Yokoi, 2015. "On the Lattice Structure of Stable Allocations in a Two-Sided Discrete-Concave Market," Mathematics of Operations Research, INFORMS, vol. 40(2), pages 460-473, February.
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