Robert W. Irving

An efficient algorithm for the "stable roommates" problem

Abstract The stable marriage problem is that of matching n men and n women, each of whom has ranked the members of the opposite sex in order of preference, so that no unmatched couple both prefer each other to their partners under the matching. At least one stable matching exists for every stable marriage instance, and efficient algorithms for finding such a matching are well known. The stable roommates problem involves a single set of even cardinality n, each member of which ranks all the others in order of preference. A stable matching is now a partition of this single set into n 2 pairs so that no two unmatched members both prefer each other to their partners under the matching. In this case, there are problem instances for which no stable matching exists. However, the present paper describes an O(n2) algorithm that will determine, for any instance of the problem, whether a stable matching exists, and if so, will find such a matching.

An efficient algorithm for the “optimal” stable marriage

In an instance of size n of the stable marriage problem, each of n men and n women ranks the members of the opposite sex in order of preference. A stable matching is a complete matching of men and women such that no man and woman who are not partners both prefer each other to their actual partners under the matching. It is well known [2] that at least one stable matching exists for every stable marriage instance. However, the classical Gale-Shapley algorithm produces a marriage that greatly favors the men at the expense of the women, or vice versa. The problem arises of finding a stable matching that is optimal under some more equitable or egalitarian criterion of optimality. This problem was posed by Knuth [6] and has remained unsolved for some time. Here, the objective of maximizing the average (or, equivalently, the total) “satisfaction” of all people is used. This objective is achieved when a persons satisfaction is measured by the position of his/her partner in his/her preference list. By exploiting the structure of the set of all stable matchings, and using graph-theoretic methods, an O(n4) algorithm for this problem is derived.

The complexity of counting stable marriages

In an instance of size n of the stable marriage problem, each of n men and n women ranks the members of the opposite sex in order of preference. A stable matching is a complete matching of men and women such that no man and woman who are not partners both prefer each other to their actual partners under the matching.It is well known that at least one stable matching exists for every stable marriage instance, so that the decision version of the problem always has a “yes” answer. Furthermore, efficient algorithms are known for the determination of such a stable matching, so that the search version of the problem is polynomially solvable.However, by exploring the structure of the set of stable matchings for any particular instance of the problem, and exploiting its relationship with the set of antichains of an associated partially ordered set, we prove that the enumeration version of the problem—determining the number of stable matchings—is # P-complete, and therefore cannot be solved in polynomial time if

Stable marriage and indifference

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The College Admissions problem with lower and common quotas

Abstract It is well known that every instance of the classical stable marriage problem admits at least one stable matching, and that such a matching can be found in O(n2) time by application of the Gale/Shapley algorithm. In the classical version of the problem, each person must rank the members of the opposite sex in strict order of preference. In practical applications, a person may not wish (or be able) to choose between alternatives, thus allowing ties in the preference lists (or, more generally, allowing each preference list to be a partial order). With the introduction of such indifference, the notion of stability may be generalised in three obvious ways. For the weakest extension of stability, the same existence result holds, and essentially the same algorithm may be applied. In the other two cases, however, there is no guarantee that stable matchings exist. Nonetheless, in this paper, we describe polynomial-time algorithms that will establish, in either of these two cases, whether a matching of the appropriate kind exists, and if so will find such a matching.

Database indexing for large DNA and protein sequence collections

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.

The Hospitals/Residents Problem with Ties

Abstract. Our aim is to develop new database technologies for the approximate matching of unstructured string data using indexes. We explore the potential of the suffix tree data structure in this context. We present a new method of building suffix trees, allowing us to build trees in excess of RAM size, which has hitherto not been possible. We show that this method performs in practice as well as the O(n) method of Ukkonen [70]. Using this method we build indexes for 200 Mb of protein and 300 Mbp of DNA, whose disk-image exceeds the available RAM. We show experimentally that suffix trees can be effectively used in approximate string matching with biological data. For a range of query lengths and error bounds the suffix tree reduces the size of the unoptimised O(mn) dynamic programming calculation required in the evaluation of string similarity, and the gain from indexing increases with index size. In the indexes we built this reduction is significant, and less than 0.3% of the expected matrix is evaluated. We detail the requirements for further database and algorithmic research to support efficient use of large suffix indexes in biological applications.

Approximability results for stable marriage problems with ties

The hospitals/residents problem is an extensively-studied many-one stable matching problem. Here, we consider the hospitals/ residents problem where ties are allowed in the preference lists. In this extended setting, a number of natural definitions for a stable matching arise. We present the first linear-time algorithm for the problem under the strongest of these criteria, so-called super-stability. Our new results have applications to large-scale matching schemes, such as the National Resident Matching Program in the US, and similar schemes elsewhere.

Sorting Strings by Reversals and by Transpositions

We consider instances of the classical stable marriage problem in which persons may include ties in their preference lists. We show that, in such a setting, strong lower bounds hold for the approximability of each of the problems of finding an egalitarian, minimum regret and sex-equal stable matching. We also consider stable marriage instances in which persons may express unacceptable partners in addition to ties. In this setting, we prove that there are constants ?,?? such that each of the problems of approximating a maximum and minimum cardinality stable matching within factors of ?,?? (respectively) is NP-hard, under strong restrictions. We also give an approximation algorithm for both problems that has a performance guarantee expressible in terms of the number of lists with ties. This significantly improves on the best-known previous performance guarantee, for the case that the ties are sparse. Our results have applications to large-scale centralized matching schemes.

On approximating the minimum independent dominating set

The problems of sorting by reversals and sorting by transpositions have been studied because of their applications to genome comparison. Prior studies of both problems have assumed that the sequences to be compared (or sorted) contain no duplicates, but there is a natural generalization in which the sequences are allowed to contain repeated characters. In this paper we study primarily the versions of these problems in which the strings to be compared are drawn from a binary alphabet. We obtain upper and lower bounds for reversal and transposition distance and show that the problem of finding reversal distance between binary strings, and therefore between strings over an arbitrary fixed-size alphabet, is NP-hard.