Size versus stability in the marriage problem

Given an instance I of the classical Stable Marriage problem with Incomplete preference lists (smi), a maximum cardinality matching can be larger than a stable matching. In many large-scale applications of smi, we seek to match as many agents as possible. This motivates the problem of finding a maximum cardinality matching in I that admits the smallest number of blocking pairs (so is “as stable as possible”). We show that this problem is NP-hard and not approximable within n1−ε, for any ε>0, unless P=NP, where n is the number of men in I. Further, even if all preference lists are of length at most 3, we show that the problem remains NP-hard and not approximable within δ, for some δ>1. By contrast, we give a polynomial-time algorithm for the case where the preference lists of one sex are of length at most 2. We also extend these results to the cases where (i) preference lists may include ties, and (ii) we seek to minimize the number of agents involved in a blocking pair

Locally Stable Marriage with Strict Preferences

We study stable matching problems with locality of information and control. In our model, each agent is a node in a fixed network and strives to be matched to another agent. An agent has a complete preference list over all other agents it can be matched with. Agents can match arbitrarily, and they learn about possible partners dynamically based on their current neighborhood. We consider convergence of dynamics to locally stable matchings -- states that are stable with respect to their imposed information structure in the network. In the two-sided case of stable marriage in which existence is guaranteed, we show that the existence of a path to stability becomes NP-hard to decide. This holds even when the network exists only among one partition of agents. In contrast, if one partition has no network and agents remember a previous match every round, a path to stability is guaranteed and random dynamics converge with probability 1. We characterize this positive result in various ways. For instance, it holds for random memory and for cache memory with the most recent partner, but not for cache memory with the best partner. Also, it is crucial which partition of the agents has memory. Finally, we present results for centralized computation of locally stable matchings, i.e., computing maximum locally stable matchings in the two-sided case and deciding existence in the roommates case.Comment: Conference version in ICALP 2013; to appear in SIAM J. Disc Mat

Socially stable matchings in the hospitals / residents problem

In the Hospitals/Residents (HR) problem, agents are partitioned into hospitals and residents. Each agent wishes to be matched to an agent in the other set and has a strict preference over these potential matches. A matching is stable if there are no blocking pairs, i.e., no pair of agents that prefer each other to their assigned matches. Such a situation is undesirable as it could lead to a deviation in which the blocking pair form a private arrangement outside the matching. This however assumes that the blocking pair have social ties or communication channels to facilitate the deviation. Relaxing the stability definition to take account of the potential lack of social ties between agents can yield larger stable matchings. In this paper, we define the Hospitals/Residents problem under Social Stability (HRSS) which takes into account social ties between agents by introducing a social network graph to the HR problem. Edges in the social network graph correspond to resident-hospital pairs in the HR instance that know one another. Pairs that do not have corresponding edges in the social network graph can belong to a matching M but they can never block M. Relative to a relaxed stability definition for HRSS, called social stability, we show that socially stable matchings can have different sizes and the problem of finding a maximum socially stable matching is NP-hard, though approximable within 3/2. Furthermore we give polynomial time algorithms for three special cases of the problem

A constraint programming approach to the hospitals/residents problem

An instance I of the Hospitals/Residents problem (HR) involves a set of residents (graduating medical students) and a set of hospitals, where each hospital has a given capacity. The residents have preferences for the hospitals, as do hospitals for residents. A solution of I is a <i>stable matching</i>, which is an assignment of residents to hospitals that respects the capacity conditions and preference lists in a precise way. In this paper we present constraint encodings for HR that give rise to important structural properties. We also present a computational study using both randomly-generated and real-world instances. We provide additional motivation for our models by indicating how side constraints can be added easily in order to solve hard variants of HR

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

A Stable Marriage Requires Communication

The Gale-Shapley algorithm for the Stable Marriage Problem is known to take $\Theta(n^2)$ steps to find a stable marriage in the worst case, but only $\Theta(n \log n)$ steps in the average case (with $n$ women and $n$ men). In 1976, Knuth asked whether the worst-case running time can be improved in a model of computation that does not require sequential access to the whole input. A partial negative answer was given by Ng and Hirschberg, who showed that $\Theta(n^2)$ queries are required in a model that allows certain natural random-access queries to the participants' preferences. A significantly more general - albeit slightly weaker - lower bound follows from Segal's general analysis of communication complexity, namely that $\Omega(n^2)$ Boolean queries are required in order to find a stable marriage, regardless of the set of allowed Boolean queries. Using a reduction to the communication complexity of the disjointness problem, we give a far simpler, yet significantly more powerful argument showing that $\Omega(n^2)$ Boolean queries of any type are indeed required for finding a stable - or even an approximately stable - marriage. Notably, unlike Segal's lower bound, our lower bound generalizes also to (A) randomized algorithms, (B) allowing arbitrary separate preprocessing of the women's preferences profile and of the men's preferences profile, (C) several variants of the basic problem, such as whether a given pair is married in every/some stable marriage, and (D) determining whether a proposed marriage is stable or far from stable. In order to analyze "approximately stable" marriages, we introduce the notion of "distance to stability" and provide an efficient algorithm for its computation

How Hard Is It to Satisfy (Almost) All Roommates?

The classic Stable Roommates problem (the non-bipartite generalization of the well-known Stable Marriage problem) asks whether there is a stable matching for a given set of agents, i.e. a partitioning of the agents into disjoint pairs such that no two agents induce a blocking pair. Herein, each agent has a preference list denoting who it prefers to have as a partner, and two agents are blocking if they prefer to be with each other rather than with their assigned partners. Since stable matchings may not be unique, we study an NP-hard optimization variant of Stable Roommates, called Egal Stable Roommates, which seeks to find a stable matching with a minimum egalitarian cost gamma, i.e. the sum of the dissatisfaction of the agents is minimum. The dissatisfaction of an agent is the number of agents that this agent prefers over its partner if it is matched; otherwise it is the length of its preference list. We also study almost stable matchings, called Min-Block-Pair Stable Roommates, which seeks to find a matching with a minimum number beta of blocking pairs. Our main result is that Egal Stable Roommates parameterized by gamma is fixed-parameter tractable, while Min-Block-Pair Stable Roommates parameterized by beta is W[1]-hard, even if the length of each preference list is at most five