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Faster Algorithms for Stable Allocation Problems

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Abstract

We consider a high-multiplicity generalization of the classical stable matching problem known as the stable allocation problem, introduced by Baïou and Balinski in 2002. By leveraging new structural properties and sophisticated data structures, we show how to solve this problem in O(mlog n) time on a bipartite instance with n vertices and m edges, improving the best known running time of O(mn). Building on this algorithm, we provide an algorithm for the non-bipartite stable allocation problem running in O(mlog n) time with high probability. Finally, we give a polynomial-time algorithm for solving the “optimal” variant of the bipartite stable allocation problem, as well as a 2-approximation algorithm for the NP-hard “optimal” variant of the non-bipartite stable allocation problem.

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References

  1. Baiou, M., Balinski, M.: Many-to-many matching: Stable polyandrous polygamy (or polygamous polyandry). Discrete Appl. Math. 101, 1–12 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Baiou, M., Balinski, M.: Erratum: The stable allocation (or ordinal transportation) problem. Math. Oper. Res. 27(4), 662–680 (2002)

    Article  MathSciNet  Google Scholar 

  3. Bansal, V., Agrawal, A., Malhotra, V.S.: Polynomial time algorithm for an optimal stable assignment with multiple partners. Theor. Comput. Sci. 379(3), 317–328 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Biró, P., Fleiner, T.: Integral stable allocation problem on graphs. Discrete Optim. 7(1–2), 64–73 (2010)

    Article  Google Scholar 

  5. Dean, B.C., Immorlica, N., Goemans, M.X.: Finite termination of “augmenting path” algorithms in the presence of irrational problem data. In: Proceedings of the 14th Annual European Symposium on Algorithms (ESA), pp. 268–279 (2006)

  6. Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Mon. 69(1), 9–14 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  7. Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum-flow problem. J. ACM 35(4), 921–940 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gusfield, D., Irving, R.W.: The Stable Marriage Problem: Structure and Algorithms. MIT Press, Cambridge (1989)

    MATH  Google Scholar 

  9. Irving, R.W.: An efficient algorithm for the “stable roommates” problem. J. Algorithms 6(4), 577–595 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  10. Irving, R.W., Leather, P., Gusfield, D.: An efficient algorithm for the “optimal” stable marriage. J. ACM 34(3), 532–543 (1987)

    MathSciNet  Google Scholar 

  11. Knuth, D.E.: Stable marriage and its relation to other combinatorial problems. In: CRM Proceedings and Lecture Notes, vol. 10. Am. Math. Soc., Providence (1997). (English translation of Marriages Stables, Les Presses de L’Université de Montréal, 1976)

    Google Scholar 

  12. Roth, A.E.: The evolution of the labor market for medical interns and residents: a case study in game theory. J. Polit. Econ. 92, 991–1016 (1984)

    Article  Google Scholar 

  13. Sleator, D.D., Tarjan, R.E.: A data structure for dynamic trees. J. Comput. Syst. Sci. 26(3), 362–391 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  14. Sleator, D.D., Tarjan, R.E.: Self-adjusting binary search trees. J. ACM 32(3), 652–686 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  15. Tan, J.J.M.: A necessary and sufficient condition for the existence of a complete stable matching. J. Algorithms 12, 154–178 (1991)

    Article  MATH  MathSciNet  Google Scholar 

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  1. School of Computing, Clemson University, Clemson, SC, USA

    Brian C. Dean & Siddharth Munshi

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  1. Brian C. Dean
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  2. Siddharth Munshi
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Correspondence to Brian C. Dean.

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Dean, B.C., Munshi, S. Faster Algorithms for Stable Allocation Problems. Algorithmica 58, 59–81 (2010). https://doi.org/10.1007/s00453-010-9416-y

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  • DOI: https://doi.org/10.1007/s00453-010-9416-y

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