Stable Roommates Research Articles

The Stable Roommates Problem with Ties

We study the variant of the well-known stable roommates problem in which participants are permitted to express ties in their preference lists. In this setting, more than one definition of stability is possible. Here we consider two of these stability criteria, so-called super-stability and weak stability. We present a linear–time algorithm for finding a super-stable matching if one exists, given a stable roommates instance with ties. This contrasts with the known NP-hardness of the analogous problem under weak stability. We also extend our algorithm to cope with preference lists that are incomplete and/or partially ordered. On the other hand, for a given stable roommates instance with ties and incomplete lists, we show that the weakly stable matchings may be of different sizes and the problem of finding a maximum cardinality weakly stable matching is NP-hard, though approximable within a factor of 2.

On the complexity of exchange-stable roommates

For the stable roommates problem recently a new concept, exchange stability, was introduced. In this paper we prove that it is NP-complete to decide whether for a given preference profile an exchange stable matching exists.

Stability in coalition formation games

In the context of coalition formation games a player evaluates a partition on the basis of the set she belongs to. For this evaluation to be possible, players are supposed to have preferences over sets to which they could belong. In this paper, we suggest two extensions of preferences over individuals to preferences over sets. For the first one, derived from the most preferred member of a set, it is shown that a strict core partition always exists if the original preferences are strict and a simple algorithm for the computation of one strict core partition is derived. This algorithm turns out to be strategy proof. The second extension, based on the least preferred member of a set, produces solutions very similar to those for the stable roommates problem.

On the Existence of Stable Roommate Matchings

This paper identifies a condition called “no odd rings” that is sufficient for the existence of stable roommate matchings in the weak preferences case. It shows that the process of allowing randomly chosen blocking pairs to match converges to a stable roommate matching with probability one as long as there are no odd rings. This random-paths-to-stability result generalizes that of Roth and Vande Vate (1990, Econometrica58, 1475–1480) and may not hold if there are odd rings. The “no odd rings” condition can also be used to prove a number of other sufficient conditions that are more economically interpretable. Journal of Economic Literature Classification Numbers: C78, D71.

On a cutting plane heuristic for the stable roommates problem and its applications

We propose a new cutting plane heuristic for the classical stable roommates problem. Our approach utilises a new linear programming formulation for the problem, and the underlying geometric properties of the fractional solution to construct the violated inequality. We test the approach on moderate size problems, with encouraging computational performance. To further illustrate the versatility of this approach, we also show how it can be suitably extended to handle variants of the basic stable roommates model.

Bistable Versions of the Marriages and Roommates Problems

A stable matching for an instance of the stable marriages problem or the stable roommates problem is bistable if it is also a stable matching when the ordering of the input preference lists is reversed. For the stable marriages problem, it is shown that the bistable matchings are a sublattice of the distributive lattice of stable matchings. In addition, the Gale–Shapley algorithm is modified to find the man-optimal bistable matching and to determine the irreducible bistable matchings.

The Geometry of Fractional Stable Matchings and Its Applications

We study the classical stable marriage and stable roommates problems using a polyhedral approach. We propose a new LP formulation for the stable roommates problem, which has a feasible solution if and only if the underlying roommates problem has a stable matching. Furthermore, for certain special weight functions on the edges, we construct a 2-approximation algorithm for the optimal stable roommates problem. Our technique exploits features of the geometry of fractional solutions of this formulation. For the stable marriage problem, we show that a related geometry allows us to express any fractional solution in the stable marriage polytope as a convex combination of stable marriage solutions. This also leads to a genuinely simple proof of the integrality of the stable marriage polytope.

A generalization of the stable matching problem

It is known that there may not exist any stable matching for a given instance of the stable roommates problem. A stable partition is a structure that generalizes the notion of a stable matching; Tan (1991) proved that every instance of the stable roommates problem contains at least one such structure. In this paper we propose a new algorithm for finding a stable partition, and hence a new algorithm for finding a stable matching if one exists. Our algorithm processes the problem dynamically as long as certain relative preference orders are maintained. Some theoretical results about stable partitions are also presented.

A New Approach to Stable Matching Problems

It is shown that Stable Matching problems are the same as problems about stable configurations of X-networks. Consequences include easy proofs of old theorems, a new simple algorithm for finding a stable matching, an understanding of the difference between Stable Marriage and Stable Roommates, NP-completeness of Three-party Stable Marriage, CC-completeness of several Stable Matching problems, and a fast parallel reduction from the Stable Marriage problem to the Assignment problem.

An upper bound for the solvability probability of a random stable roommates instance

AbstractIt is well‐known that not all instances of the stable roommates problem admit a stable matching. Here we establish the first nontrivial upper bound on the limiting behavior of Pn, the probability that a random roommates instance of size n has a stable matching, namely, limn→∞ Pn⩽ e1/2/2 (=0.8244…). © 1994 John Wiley & Sons, Inc.

The "Stable Roommates" Problem with Random Preferences

In a set of even cardinality $n$, each member ranks all the others in order of preference. A stable matching is a partition of the set into $n/2$ pairs, with the property that no two unpaired members both prefer each other to their partners under matching. It is known that for some problem instances no stable matching exists. What if an instance of the ranking system is chosen uniformly at random? We show that the mean and the variance of the total number of stable matchings for the random problem instance are asymptotic to $e^{1/2}$ and $(\pi n/4e)^{1/2}$, respectively. Consequently, $\operatorname{Prob}$ a stable matching exists) $\gtrsim (4e^3/\pi n)^{1/2}$. We also prove that, given the last event, in every stable matching the sum of the ranks of all members (as rank ordered by their partners) is asymptotic to $n^{3/2}$, and the largest rank of a partner is of order $n^{1/2}\log n$, with superpolynomially high conditional probability. In other words, stable partners are very likely to be relatively close to the tops of each other's preference lists.

On a Random Instance of a ‘Stable Roommates’ Problem: Likely Behavior of the Proposal Algorithm

In a set of even cardinalityn, each member ranks all the others in order of preference. A stable matching is a partition of the set inton/2 pairs, with the property that no two unpaired members both prefer each other to their partners under matching. It is known that for some problem instances no stable matching exists. In 1985, Irving found anO(n2) two-phase algorithm that would determine, for any instance, whether a stable matching exists, and if so, would find such a matching. Recently, Tan proved that Irving's algorithm, with a modified second phase, always finds a stable cyclic partition of the members set, which is a stable matching when each cycle has length two. In this paper we study a likely behavior of the algorithm under the assumption that an instance of the ranking system is chosen uniformly at random. We prove that the likely number of basic steps,i.e.the individual proposals in the first phase and the rotation eliminations, involving subsets of members in the second phase, isO(nlogn), and that the likely size of a rotation isO((nlogn)1/2). We establish a ‘hyperbola law’ analogous to our past result on stable marriages. It states that at every step of the second phase, the product of the rank of proposers and the rank of proposal holders is asymptotic, in probability, ton3. We show that every stable cyclic partition is likely to be almost a stable matching, in the sense that at mostO((nlogn)1/2) members can be involved in the cycles of length three or more.

A bounded approximation for the minimum cost 2-sat problem

Given a satisfiable Boolean formula in 2-CNF, it is NP-hard to find a satisfying assignment that contains a minimum number of true variables. A polynomial-time approximation algorithm is given that finds an assignment with at most twice as many true variables as necessary. The algorithm also works for a weighted generalization of the problem. An application to the optimal stable roommates problem is given in detail, and other applications are mentioned.

A new fixed point approach for stable networks and stable marriages

In a network stability problem, the aim is to find stable configurations for a given network of boolean gates. For general networks, the problem is known to be computationally hard. Mayr and Subramanian introduced an interesting class of networks by imposing fanout restrictions at each gate, and showed that network stability on this class of networks is still sufficiently rich to express as special cases stable marriage and stable roommate problems. In this paper we study the sequential and parallel complexity of network stability on networks with restricted fanout. Our approach builds on structural properties of these networks and exposes close ties with the theory of retracts and isometric embeddings for product graphs. This structure then gives new efficient algorithms for questions of representation, enumeration, and optimization in stable matching.

The complexity of circuit value and network stability

We develop a method for nontrivially restricting fanout in a circuit. We study the complexity of the circuit value problem and a new problem, network stability, when fanout is limited. This leads to new classes of problems within P . We conjecture that the new classes are different from P and incomparable to NC . One of these classes, CC . contains several natural complete problems, including circuit value for comparator circuits, lex-first maximal matching, and problems related to stable marriage and stable roommates. When fanout is appropriately limited, we obtain positive results: a parallel algorithm for circuit value that runs in time about the square root of the number of gates, a linear-time sequential algorithm for network stability, and logspace reductions between circuit value and network stability.

On finding a stable roommate, job, or spouse

The use of real world problems as the basis for assignments in Computer Science courses is attractive for many reasons. At the same time it is difficult to find such a problem that offers the same richness that is found, for example, in sorting or searching. In this paper a problem is presented that has many real-world instances and which is pedagogically attractive at all levels of Computer Science from the level of a non-major's course to that of an advanced algorithms course.

Three-Dimensional Stabl Matching Problems

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, the complexity of three-dimensional extensions of these problems is investigated. This is one of twelve research directions suggested by Knuth in his book on the stable marriage problem. It is shown that these problems are NP-complete, and hence it is unlikely that there exist efficient algorithms for their solutions. The approach developed in this paper provides an alternate NP-completeness proof for the hospitals/residents problem with couples—an important practical problem shown earlier to be NP-complete by Ronn.

The Stable Marriage Problem: Structure and Algorithms

This book probes the stable marriage problem and its variants as a rich source of problems and ideas that illustrate both the design and analysis of efficient algorithms. It covers the most recent structural and algorithmic work on stable matching problems, simplifies and unifies many earlier proofs, strengthens several earlier results, and presents new results and more efficient algorithms.The authors develop the structure of the set of stable matchings in the stable marriage problem in a more general and algebraic context than has been done previously; they discuss the problem's structure in terms of rings of sets, which allows many of the most useful features to be seen as features of a more general set of problems. The relationship between the structure of the stable marriage problem and the more general stable roommates problem is demonstrated, revealing many commonalities.The results the authors obtain provide an algorithmic response to the practical, and political, problems created by the asymmetry inherent in the Gale Shapley solutions, leading to alternative methods and better compromises than are provided by the Gale Shapley method. And, in contrast to Donald Knuth's earlier work which primarily focused on the application of mathematics to the analysis of algorithms, this book illustrates the productive and almost inseparable relationship between mathematical insight and the design of efficient algorithms.Dan Gusfield is Associate Professor of Computer Science at the University of California, Davis. Robert W. Irving is Senior Lecturer in Computing Science at the University of Glasgow. The Stable Marriage Problem is included in the Foundations of Computing Series, edited by Michael Garey and Albert Meyer.

A necessary and sufficient condition for the existence of a complete stable matching

The stable roommates problem is a well-known problem of matching n people into n 2 disjoint pairs so that no two unmatched persons both prefer each other to their partners under the matching. We call such a matching “ a complete stable matching.” It is known that a complete stable matching may not exist. Irving described an O( n 2) algorithm that would find one complete stable matching if there is one, or would report that none exists. In this paper, we give a necessary and sufficient condition for the existence of a complete stable matching; namely, the non-existence of any odd party, which will be defined subsequently. We define a new structure called a “stable partition,” which generalizes the notion of a complete stable matching, and prove that every instance of the stable roommates problem has at least one such structure. We also show that a stable partition contains all the odd parties, if there are any. Finally we have an O( n 2) algorithm that finds one stable partition which in turn gives all the odd parties.

A unifying approach to the structures of the stable matching problems

It is well-known that the structure of the set of stable marriages of a stable marriage instance can be represented as a finite distributive lattice and, conversely, every finite distributive lattice is a set of stable marriages for some stable marriage instance. Recently, Irving [12] and Gusfield [9] propose some representations of the set of all stable assignments for a given solvable instance of the stable roommates problem. In this paper, we will give a unifying approach to the structures of the stable marriage problem and the stable roommates problem. To achieve this purpose, we first study the duality in the structure of a stable marriage instance, then transform every stable roommates instance into a corresponding stable marriage instance and obtain the structure of the stable roommates instance directly from that of the corresponding stable marriage instance. The main results of this paper are: (1) There is a one-one correspondence between the set of stable marriages for a stable marriage instance and the set of feasible words of some Faigle geometry; (2) There is a one-one correspondence between the set of stable assignments for a stable roommates instance and the set of basic words of some Faigle geometry.