The Complexity of Approximately Counting Stable Matchings

arXiv:1004.1836v4 [cs.CC] 7 Feb 2012 The Complexity of Approximately Counting Stable Matchings Prasad Chebolu∗† Leslie Ann Goldberg∗ Russell Martin∗† Abstract We investigate the complexity of approximately counting stable matchings in the k-attribute model, where the preference lists are determined by dot products of “pref- erence vectors” with “attribute vectors”, or by Euclidean distances between “prefer- ence points“ and “attribute points”. Irving and Leather [16] proved that counting the number of stable matchings in the general case is #P-complete. Counting the number of stable matchings is reducible to counting the number of downsets in a (re- lated) partial order [16] and is interreducible, in an approximation-preserving sense, to a class of problems that includes counting the number of independent sets in a bipartite graph (#BIS) [7]. It is conjectured that no FPRAS exists for this class of problems. We show this approximation-preserving interreducibilty remains even in the restricted k-attribute setting when k ≥3 (dot products) or k ≥2 (Euclidean distances). Finally, we show it is easy to count the number of stable matchings in the 1-attribute dot-product setting. 1 Introduction 1.1 Stable Matchings The stable matching problem (or stable marriage problem) is a classical combinatorics prob- lem. An instance of this problem consists of n men and n women, where each man has his own preference list (a total ordering) of the women, and, similarly, each woman has her own preference list of the men. A one-to-one pairing of the men with the women is called a matching (or marriage). Given a matching, if there exists a man M and a woman w in the matching who prefer each other over their partners in the matching, then the matching is A preliminary version of this paper appeared in APPROX 2010, Lecture Notes in Computer Science 6302, Springer, pp. 81–94. ∗Department of Computer Science, Ashton Bldg, Ashton St, University of Liverpool, Liverpool L69 3BX, United Kingdom. †Research supported in part by EPSRC Grant EP/F020651/1. 1 considered unstable and the man-woman pair (M, w) is called a blocking pair. (M and w would prefer to drop their current partners and pair up with each other.) If a matching has no blocking pairs, then we call it a stable matching. In 1962, Gale and Shapley proved that every stable matching instance has a stable matching, and described an O(n2) algorithm for finding one [8]. The stable matching problem has many variants, where ties in the preference lists could be allowed, where people might have partial preference lists (i.e. someone might prefer to remain single rather than be paired with certain members of the opposite sex), generalizations to men/women/pets, universities and applicants, students and projects, etc. Some of these generalizations have also been well-studied and, indeed, algorithms for finding stable matchings are used for assigning residents to hospitals in Scotland, Canada, and the USA [4, 20, 22]. In this paper, we concentrate solely on the classical problem, so the term “matching instance” will refer to one where the number of men is equal to the number of women, and each man or women has their own full totally-ordered (i.e. no ties allowed) preference list for the opposite sex. Irving and Leather [16] demonstrated that counting the number of stable matchings for a given instance is #P-complete. This completeness result relies on the connection between stable marriages and downsets in a related partial order (explained in more detail in Section 3), as counting the number of downsets in a partial order is another classical #P-complete problem [21]. Knowing that exactly counting stable matchings is difficult (under standard complexity- theoretic assumptions), one might turn to methods for approximately counting this number. In particular, we would like to find a fully-polynomial randomized approximation scheme (an FPRAS) for this task, i.e. an algorithm that provides an arbitrarily close approxi- mation in time polynomial in the input size and the desired error — see Section 2 for a formal definition. One method that has proven successful for other counting problems is the Markov Chain Monte Carlo (MCMC) method. This technique exploits a relationship between counting and sampling described by Jerrum, Valiant, and Vazirani [17], namely, for self-reducible combinatorial structures, the existence of an FPRAS is computationally equivalent to a polynomial-time algorithm for approximate sampling from the set of struc- tures. Although the set of stable matchings for an instance does not obviously fit into the class of self-reducible problems, an efficient algorithm for (approximately) sampling a random stable matching can be transformed into a method for (approximately) counting this number. Bhatnagar, Greenberg, and Randall [1] considered this problem of sampling a random stable matching using the MCMC method. They examined a natural Markov chain that uses “male-improving” and “fem

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