Eric McDermid

A 3/2-Approximation Algorithm for General Stable Marriage

In an instance of the stable marriage problem with ties and incomplete preference lists, stable matchings can have different sizes. It is APX-hard to compute a maximum cardinality stable matching, but there have recently been proposed polynomial-time approximation algorithms, with constant performance guarantees for both the general version of this problem, and for several special cases. Our contribution is to describe a

Keeping partners together: algorithmic results for the hospitals/residents problem with couples

\frac{3}{2}

Three-Sided Stable Matchings with Cyclic Preferences

-approximation algorithm for the general version of this problem, improving upon the recent

Matching with sizes (or scheduling with processing set restrictions)

\frac{5}{3}

Popular Matchings: Structure and Algorithms

-approximation algorithm of Kiraly. Interest in such algorithms arises because of the problems application to centralized matching schemes, the best known of which involve the assignment of graduating medical students to hospitals in various countries.

A unified approach to finding good stable matchings in the hospitals/residents setting

The Hospitals/Residents problem with Couples (HRC) is a generalisation of the classical Hospitals/Residents problem (HR) that is important in practical applications because it models the case where couples submit joint preference lists over pairs of hospitals (hi,hj). We consider a natural restriction of HRC in which the members of a couple have individual preference lists over hospitals, and the joint preference list of the couple is consistent with these individual lists in a precise sense. We give an appropriate stability definition and show that, in this context, the problem of deciding whether a stable matching exists is NP-complete, even if each resident’s preference list has length at most 3 and each hospital has capacity at most 2. However, with respect to classical (Gale-Shapley) stability, we give a linear-time algorithm to find a stable matching or report that none exists, regardless of the preference list lengths or the hospital capacities. Finally, for an alternative formulation of our restriction of HRC, which we call the Hospitals/Residents problem with Sizes (HRS), we give a linear-time algorithm that always finds a stable matching for the case that hospital preference lists are of length at most 2, and where hospital capacities can be arbitrary.

Hardness results on the man-exchange stable marriage problem with short preference lists

Knuth (Mariages Stables, Les Presses de L’Université de Montréal, 1976) asked whether the stable matching problem can be generalised to three dimensions, e.g., for families containing a man, a woman and a dog. Subsequently, several authors considered the three-sided stable matching problem with cyclic preferences, where men care only about women, women only about dogs, and dogs only about men. In this paper we prove that if the preference lists may be incomplete, then the problem of deciding whether a stable matching exists, given an instance of the three-sided stable matching problem with cyclic preferences, is NP-complete. Considering an alternative stability criterion, strong stability, we show that the problem is NP-complete even for complete lists. These problems can be regarded as special types of stable exchange problems, therefore these results have relevance in some real applications, such as kidney exchange programs.

Sex-Equal Stable Matchings: Complexity and Exact Algorithms

Matching problems on bipartite graphs where the entities on one side may have different sizes are intimately related to scheduling problems with processing set restrictions. We survey the close relationship between these two problems, and give new approximation algorithms for the (NP-hard) variations of the problems in which the sizes of the jobs are restricted. Specifically, we give an approximation algorithm with an additive error of one when the sizes of the jobs are either 1 or 2, and generalise this to an approximation algorithm with an additive error of 2^k-1 for the case where each job has a size taken from the set {1,2,4,...,2^k} (for any constant integer k). We show that the above two problems become polynomial-time solvable if the processing sets are nested.

A General Reduction Theorem with Applications to Pathwidth and the Complexity of MAX 2-CSP

An instance of the popular matching problem (POP-M) consists of a set of applicants and a set of posts. Each applicant has a preference list that strictly ranks a subset of the posts. A matching M of applicants to posts is popular if there is no other matching M *** such that more applicants prefer M *** to M than prefer M to M ***. This paper provides a characterization of the set of popular matchings for an arbitrary POP-M instance in terms of a structure called the switching graph , a directed graph computable in linear time from the preference lists. We show that the switching graph can be exploited to yield efficient algorithms for a range of associated problems, including the counting and enumeration of the set of popular matchings and computing popular matchings that satisfy various additional optimality criteria. Our algorithms for computing such optimal popular matchings improve those described in a recent paper by Kavitha and Nasre [5].

Center stable matchings and centers of cover graphs of distributive lattices

The hospitals/residents (HR) problem is a many-to-one generalization of the stable marriage (SM) problem. Researchers have been interested in variants of stable matchings that either satisfy a set of additional contraints or are optimal with respect to some cost function. In this paper, we show that broad classes of feasibility and optimization stable matching problems in the HR setting can be solved efficiently provided certain tasks (such as checking the feasibility of a stable matching or computing the cost of a stable matching) can also be done efficiently. To prove our results, we make use of an HR instances meta-rotation poset to explore its stable matchings. An algorithm that can discover all the meta-rotations of the instance serves as a starting point for all our algorithms.