Hard variants of stable marriage

Abstract

The Stable Marriage Problem and its many variants have been widely studied in the literature [6, 22, 15], partly because of the inherent appeal of the problem, partly because of the elegance of the associated structures and algorithms, and partly because of important practical applications, such as the National Resident Matching Program [20] and similar large-scale matching schemes. Here, we present the first comprehensive study of variants of the problem in which the preference lists of the participants are not necessarily complete and not necessarily totally ordered. We show that, under surprisingly restrictive assumptions, a number of these variants are hard, and hard to approximate. The key observation is that, in contrast to the case where preference lists are complete or strictly ordered (or both), a given problem instance may admit stable matchings of different sizes. In this setting, examples of problems that are hard are: finding a stable matching of maximum or minimum size, determining whether a given pair is stable-even if the indifference takes the form of ties on one side only, the ties are at the tails of lists, there is at most one tie per list, and each tie is of length 2; and finding, or approximating, both an 'egalitarian' and a 'minimum regret' stable matching. However, we give a 2-approximation algorithm for the problems of finding a stable matching of maximum or minimum size. We also discuss the significant implications of our results for practical matching schemes.

The Stable Roommates Problem with Short Lists

“…An instance of SMTI always admits a stable solution and it can be found in linear time [22]. Thus these components contribute no blocking edge.…”

Modeling Stable Matching Problems with Answer Set Programming

“…In [18] it is shown that the decision problem 'is the pair (m, w) stable?' for a given SMP instance with unacceptablity and ties is an NP-complete problem, even in the absence of unacceptability.…”

“…as few or as many singles as possible. Examples of practical applications include an efficient kidney exchange program [21] and the National Resident Matching Program 3 [18]. Maximizing cardinality garantuees that as many donors as possible will get a compatible donor and as many medical graduates as possible will get a position.…”

“…Such a man and woman are called a blocking pair and a matching without blocking pairs forms a stable set of marriages. Due to its practical relevance, countless variants on the SMP have been investigated, making the problem assumptions more applicable to a wider range of applications, such as kidney-exchange [12] and the hospital-resident problem [18]. Recently Roth and Shapley won the Nobel Prize for Economics for their theory of stable allocations and the practice of market design, work that has directly resulted from an application of the SMP.…”

A Specialised Binary Constraint for the Stable Marriage Problem

“…Stable matching problems occur naturally while matching people to posts [18], such as the allocation of residents to hospitals in the US [17], Canada [5], and Scotland [11]. Variants of the problem occur such as the allocation of groups of students to university accommodation [12], and hard variants have been identified and studied by Irving and Manlove [16]. The problem has also attracted the interest of the constraint programming community [4,7,9,8,14].…”