Finding a Stable Allocation in Polymatroid Intersection

The stable matching (or stable marriage) model of Gale and Shapley (1962) has been generalized in various directions such as matroid kernels due to Fleiner (2001) and stable allocations in bipartite networks due to Bäıou and Balinski (2002). Unifying these generalizations, we introduce the concept of stable allocations in polymatroid intersection. Our framework includes both integerand real-variable versions. The integer-variable version corresponds to a special case of the discrete-concave function model due to Eguchi, Fujishige, and Tamura (2003), 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(nγ), where n and γ respectively denote the cardinality of the ground set and the time for computing the saturation and exchange capacities. 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. ∗An earlier version of this paper appeared in Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, Philadelphia, 2016, pp. 1034–1047. †Department of Mathematical Informatics, University of Tokyo, Tokyo 113-8656, Japan. E-mail: {iwata, yu yokoi}@mist.i.u-tokyo.ac.jp