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Improved algorithmic results for unsplittable stable allocation problems

Abstract

The stable allocation problem is a many-to-many generalization of the well-known stable marriage problem, where we seek a bipartite assignment between, say, jobs (of varying sizes) and machines (of varying capacities) that is “stable” based on a set of underlying preference lists submitted by the jobs and machines. Building on the initial work of Dean et al. (The unsplittable stable marriage problem, 2006), we study a natural “unsplittable” variant of this problem, where each assigned job must be fully assigned to a single machine. Such unsplittable bipartite assignment problems generally tend to be NP-hard, including previously-proposed variants of the unsplittable stable allocation problem (McDermid and Manlove in J Comb Optim 19(3): 279–303, 2010). Our main result is to show that under an alternative model of stability, the unsplittable stable allocation problem becomes solvable in polynomial time; although this model is less likely to admit feasible solutions than the model proposed in McDermid and Manlove (J Comb Optim 19(3): 279–303, McDermid and Manlove 2010), we show that in the event there is no feasible solution, our approach computes a solution of minimal total congestion (overfilling of all machines collectively beyond their capacities). We also describe a technique for rounding the solution of a stable allocation problem to produce “relaxed” unsplit solutions that are only mildly infeasible, where each machine is overcongested by at most a single job.

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Notes

  1. 1.

    The model introduced in Dean et al. (2006) allows \(x(m) \le q(m)\), but we believe strict inequality is actually a better choice—mathematically and from a modeling perspective. For example, the old definition applied to a hospital–resident matching scenario with married couples might cause a hospital to accept two more residents than its quota, while the new definition would only require accepting one more resident. The results in Dean et al. (2006) hold with either definition.

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Acknowledgments

This work was partially supported by the Deutsche Telekom Stiftung, the Deutsche Forschungsgemeinschaft within the research training group Methods for Discrete Structures (GRK 1408), and by the USA National Science Foundation CAREER award CCF-0845593.

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Correspondence to Ágnes Cseh.

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Cseh, Á., Dean, B.C. Improved algorithmic results for unsplittable stable allocation problems. J Comb Optim 32, 657–671 (2016). https://doi.org/10.1007/s10878-015-9889-3

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Keywords

  • Stable matchings
  • Stable allocations
  • Rotations
  • Unsplittable assignments