The exchange-stable marriage problem

In this paper we consider instances of stable matching problems, namely the classical stable marriage (SM) and stable roommates (SR) problems and their variants. In such instances we consider a stability criterion that has recently been proposed, that of <i>exchange-stability</i>. In particular, we prove that ESM — the problem of deciding, given an SM instance, whether an exchange-stable matching exists — is NP-complete. This result is in marked contrast with Gale and Shapley's classical linear-time algorithm for finding a stable matching in an instance of SM. We also extend the result for ESM to the SR case. Finally, we study some variants of ESM under weaker forms of exchange-stability, presenting both polynomial-time solvability and NP-completeness results for the corresponding existence questions

The College Admissions problem with lower and common quotas

We study two generalised stable matching problems motivated by the current matching scheme used in the higher education sector in Hungary. The first problem is an extension of the College Admissions problem in which the colleges have lower quotas as well as the normal upper quotas. Here, we show that a stable matching may not exist and we prove that the problem of determining whether one does is NP-complete in general. The second problem is a different extension in which, as usual, individual colleges have upper quotas, but, in addition, certain bounded subsets of colleges have common quotas smaller than the sum of their individual quotas. Again, we show that a stable matching may not exist and the related decision problem is NP-complete. On the other hand, we prove that, when the bounded sets form a nested set system, a stable matching can be found by generalising, in non-trivial ways, both the applicant-oriented and college-oriented versions of the classical Gale–Shapley algorithm. Finally, we present an alternative view of this nested case using the concept of choice functions, and with the aid of a matroid model we establish some interesting structural results for this case

Complexity of the stable marriage and stable roommate problems in three dimensions

The stable marriage problem is a matching problem that pairs members of two sets. The objective is to achieve a matching that satisfies all participants based on their preferences. The stable roommate problem is a variant involving only one set, which is partitioned into pairs with a similar objective. There exist asymptotically optimal algorithms that solve both problems.In this paper, we investigate the complexity of three dimensional extensions of these problems. This is one of twelve research directions suggested by Knuth in his book on the stable marriage problem. We show that these problems are NP-complete, and hence it is unlikely that there exist efficient algorithms for their solutions.Applying the polynomial tranformation developed in this paper, we extend the NP-completeness result to include the problem of matching couples - who are both medical school graduates - to pairs of hospital resident positions. This problem is important in practice and is dealth with annually by NRMP, the centralized program that matches all medical school graduates in the United States to available resident positions

Algorithmic aspects of stable matching problems

The Stable Marriage problem (SM), the Hospitals/Residents problem (HR) and the Stable Roommates problem (SR) are three classical stable matching problems that were first studied by Gale and Shapley in 1962. These problems have widespread practical application in centralised automated matching schemes, which assign applicants to posts based on preference lists and capacity constraints in both the UK and internationally. Within such schemes it is often the case that an agent's preference list may be incomplete, and agents may also be allowed to express indifference in the form of ties. In the presence of ties, three stability criteria can be defined, namely weak stability, strong stability and super-stability. In this thesis we consider stable matching problems from an algorithmic point of view. Some of the problems that we consider are derived from new stable matching models, whilst others are obtained from existing stable matching models involving ties and incomplete lists, with additional natural restrictions on the problem instance. Furthermore, we also explore the use of constraint programming with both SM and HR. We first study a new variant of the Student-Project Allocation problem in which each student ranks a set of acceptable projects in preference order and similarly each lecturer ranks his available projects in preference order. In this context, two stability definitions can be identified, namely weak stability and strong stability. We show that the problem of finding a maximum weakly stable matching is NP-hard. However, we describe two 2-approximation algorithms for this problem. Regarding strong stability, we describe a polynomial-time algorithm for finding such a matching or reporting that none exists. Next we investigate SM with ties and incomplete lists (SMTI), and HR with ties (HRT), where the length of each agent's list is subject to an upper bound. We present both polynomial-time algorithms and NP-hardness results for a range of problems that are derived from imposing upper bounds on the length of the lists on one or both sides. We also consider HRT, and SR with ties and incomplete lists (SRTI), where the preference lists of one or both sets of agents (as applicable) are derived from one or two master lists in which agents are ranked. For super-stability, in the case of each of HRT and SRTI with a master list, we describe a linear-time algorithm that simplifies the algorithm used in the general case. In the case of strong stability, for each of HRT and SRTI with a master list, we describe an algorithm that is faster than that for the general case. We also show that, given an instance I of SRTI with a master list, the problem of finding a weakly stable matching is polynomial-time solvable. However, we show that given such an I, the problem of finding a maximum weakly stable matching is NP-hard. Other new stable matching models that we study are the variants of SMTI and SRTI with symmetric preferences. In this context we consider two models that are derived from alternative ways of interpreting the rank of an agent in the presence of ties. For both models we show that deciding if a complete weakly stable matching exists is NP-complete. Then for one of the models we show that each of the problem of finding a minimum regret and an egalitarian weakly stable matching is NP-hard and that the problem of determining if a (man,woman) pair belongs to a weakly stable matching is NP-complete. We then describe algorithms for each of the problems of finding a super-stable matching and a strongly stable matching, or reporting that none exists, given instances of SRTI and HRT with symmetric preferences (regardless of how the ranks are interpreted). Finally, we use constraint programming techniques to model instances of SM and HR. We describe two encodings of SM in terms of a constraint satisfaction problem. The first model for SM is then extended to the case of HR. This encoding for HR is then extended to create a model for HRT under weak stability. Using this encoding we can obtain, with the aid of search, all the weakly stable matchings, given an instance of HRT

Classified Stable Matching

We introduce the {\sc classified stable matching} problem, a problem motivated by academic hiring. Suppose that a number of institutes are hiring faculty members from a pool of applicants. Both institutes and applicants have preferences over the other side. An institute classifies the applicants based on their research areas (or any other criterion), and, for each class, it sets a lower bound and an upper bound on the number of applicants it would hire in that class. The objective is to find a stable matching from which no group of participants has reason to deviate. Moreover, the matching should respect the upper/lower bounds of the classes. In the first part of the paper, we study classified stable matching problems whose classifications belong to a fixed set of ``order types.'' We show that if the set consists entirely of downward forests, there is a polynomial-time algorithm; otherwise, it is NP-complete to decide the existence of a stable matching. In the second part, we investigate the problem using a polyhedral approach. Suppose that all classifications are laminar families and there is no lower bound. We propose a set of linear inequalities to describe stable matching polytope and prove that it is integral. This integrality allows us to find various optimal stable matchings using Ellipsoid algorithm. A further ramification of our result is the description of the stable matching polytope for the many-to-many (unclassified) stable matching problem. This answers an open question posed by Sethuraman, Teo and Qian

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