Stable Marriage with Ties and Bounded Length Preference Lists

We consider variants of the classical stable marriage problem in which preference lists may contain ties, and may be of bounded length. Such restrictions arise naturally in practical applications, such as centralised matching schemes that assign graduating medical students to their first hospital posts. In such a setting, weak stability is the most common solution concept, and it is known that weakly stable matchings can have different sizes. This motivates the problem of finding a maximum cardinality weakly stable matching, which is known to be NP-hard in general. We show that this problem is solvable in polynomial time if each man's list is of length at most 2 (even for women's lists that are of unbounded length). However if each man's list is of length at most 3, we show that the problem becomes NP-hard and not approximable within some d > 1, even if each woman's list is of length at most 4

An 8/5 approximation algorithm for a hard variant of stable marriage

When ties and incomplete preference lists are permitted in the Stable Marriage problem, stable matchings can have different sizes. The problem of finding a maximum cardinality stable matching in this context is NP-hard, even under very severe restrictions on the number, size and position of ties. In this paper, we describe a polynomial-time 8/5-approximation algorithm for a variant in which ties are on one side only and at the end of the preference lists. This variant is motivated by important applications in large scale centralized matching schemes

Stable marriage with ties and bounded length preference lists

We consider variants of the classical stable marriage problem in which preference lists may contain ties, and may be of bounded length. Such restrictions arise naturally in practical applications, such as centralised matching schemes that assign graduating medical students to their first hospital posts. In such a setting, weak stability is the most common solution concept, and it is known that weakly stable matchings can have different sizes. This motivates the problem of finding a maximum cardinality weakly stable matching, which is known to be NP-hard in general. We show that this problem is solvable in polynomial time if each man's list is of length at most 2 (even for women's lists that are of unbounded length). However if each man's list is of length at most 3, we show that the problem becomes NP-hard (even if each women's list is of length at most 3) and not approximable within some δ>1 (even if each woman's list is of length at most 4)

An Algorithm for Stable Matching with Approximation up to the Integrality Gap

In the stable matching problem we are given a bipartite graph G = (A ∪ B, E) where A and B represent disjoint groups of agents, each of whom has ordinal preferences over the members of the opposite group. The goal is to find an assignment of agents in one group to those in the other such that no pair of agents prefer each other to their assignees. In this thesis we study the stable matching problem with ties and incomplete preferences. If agents are allowed to have ties and incomplete preferences, computing a stable matching of maximum cardinality is known to be NP-hard. Furthermore, it is known to be NP-hard to achieve a performance guarantee of 33/29 − ε (≈ 1.1379) and UGC-hard to attain that of 4/3 − ε (≈ 1.3333). We present a polynomial-time approximation algorithm with a performance guarantee of (3L − 2)/(2L − 1) where L is the maximum tie length. Our result matches the known lower bound on the integrality gap for the associated LP formulation

A structural approach to matching problems with preferences

This thesis is a study of a number of matching problems that seek to match together pairs or groups of agents subject to the preferences of some or all of the agents. We present a number of new algorithmic results for five specific problem domains. Each of these results is derived with the aid of some structural properties implicitly embedded in the problem. We begin by describing an approximation algorithm for the problem of finding a maximum stable matching for an instance of the stable marriage problem with ties and incomplete lists (MAX-SMTI). Our polynomial time approximation algorithm provides a performance guarantee of 3/2 for the general version of MAX-SMTI, improving upon the previous best approximation algorithm, which gave a performance guarantee of 5/3. Next, we study the sex-equal stable marriage problem (SESM). We show that SESM is W[1]-hard, even if the men's and women's preference lists are both of length at most three. This improves upon the previously known hardness results. We contrast this with an exact, low-order exponential time algorithm. This is the first non-trivial exponential time algorithm known for this problem, or indeed for any hard stable matching problem. Turning our attention to the hospitals / residents problem with couples (HRC), we show that HRC is NP-complete, even if very severe restrictions are placed on the input. By contrast, we give a linear-time algorithm to find a stable matching with couples (or report that none exists) when stability is defined in terms of the classical Gale-Shapley concept. This result represents the most general polynomial time solvable restriction of HRC that we are aware of. We then explore the three dimensional stable matching problem (3DSM), in which we seek to find stable matchings across three sets of agents, rather than two (as in the classical case). We show that under two natural definitions of stability, finding a stable matching for a 3DSM instance is NP-complete. These hardness results resolve some open questions in the literature. Finally, we study the popular matching problem (POP-M) in the context of matching a set of applicants to a set of posts. We provide a characterization of the set of popular matchings for an arbitrary POP-M instance in terms of a new structure called the switching graph exploited to yield efficient algorithms for a range of associated problems, extending and improving upon the previously best-known results for this problem

Randomized approximation of the stable marriage problem

AbstractWhile the original stable marriage problem requires all participants to rank all members of the opposite sex in a strict order, two natural variations are to allow for incomplete preference lists and ties in the preferences. Either variation is polynomially solvable, but it has recently been shown to be NP-hard to find a maximum cardinality stable matching when both of the variations are allowed. It is easy to see that the size of any two stable matchings differ by at most a factor of two, and so, an approximation algorithm with a factor two is trivial. In this paper, we give a randomized approximation algorithm RANDBRK and show that its expected approximation ratio is at most 10/7(<1.4286) for a restricted but still NP-hard case, where ties occur in only men's lists, each man writes at most one tie, and the length of ties is two. We also show that our analysis is nearly tight by giving a lower bound 32/23(>1.3913) for RANDBRK. Furthermore, we show that these restrictions except for the last one can be removed without increasing the approximation ratio too much