We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose–accept rounds executed by the Gale–Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about the problem instance; for each participant, knowing only its local neighbourhood is enough. In distributed-systems parlance, this means that if each person has only a constant number of acceptable partners, an almost stable matching emerges after a constant number of synchronous communication rounds. We apply our results to give a distributed (2+e)-approximation algorithm for maximum-weight matching in bicoloured graphs and a centralised randomised constant-time approximation scheme for estimating the size of a stable matching.
Abraham al @cite_6 study almost stable matchings in the stable roommates problem. The recent work by Bir al @cite_28 is particularly close to ours: they, too, consider the stable marriage problem with incomplete preference lists, and aim at finding a matching with few unstable edges. However, in terms of computational complexity, their work goes in the opposite direction. Their task is to find a matching that minimises the number of unstable edges. It turns out that this makes the problem computationally much more : the problem is NP-hard, unlike the classical stable marriage problem. In contrast, we do not require that the matching is a maximum matching, which makes the problem computationally : the problem admits a constant-time distributed algorithm, unlike the classical stable marriage problem. The algorithm works even when ties in the preferences lists are allowed; this should be contrasted with the fact that if ties are allowed, it is NP-hard to find a stable perfect matching @cite_30 @cite_24 @cite_11 .
@cite_30 @cite_28 @cite_6 @cite_24 @cite_11
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An instance of the classical Stable Roommates problem (sr) need not admit a stable matching. This motivates the problem of finding a matching that is “as stable as possible”, i.e. admits the fewest number of blocking pairs. In this paper we prove that, given an sr instance with n agents, in which all preference lists are complete, the problem of finding a matching with the fewest number of blocking pairs is NP-hard and not approximable within @math , for any e>0, unless P=NP. If the preference lists contain ties, we improve this result to n1−e. Also, we show that, given an integer K and an sr instance I in which all preference lists are complete, the problem of deciding whether I admits a matching with exactly K blocking pairs is NP-complete. By contrast, if K is constant, we give a polynomial-time algorithm that finds a matching with at most (or exactly) K blocking pairs, or reports that no such matching exists. Finally, we give upper and lower bounds for the minimum number of blocking pairs over all matchings in terms of some properties of a stable partition, given an sr instance I. We consider variants of the classical stable marriage problem in which preference lists may contain ties, and may be of bounded length. Such restrictions arise naturally in practical applications, such as centralised matching schemes that assign graduating medical students to their first hospital posts. In such a setting, weak stability is the most common solution concept, and it is known that weakly stable matchings can have different sizes. This motivates the problem of finding a maximum cardinality weakly stable matching, which is known to be NP-hard in general. We show that this problem is solvable in polynomial time if each man's list is of length at most 2 (even for women's lists that are of unbounded length). However if each man's list is of length at most 3, we show that the problem becomes NP-hard (even if each women's list is of length at most 3) and not approximable within some @d>1 (even if each woman's list is of length at most 4). The original stable marriage problem requires all men and women to submit a complete and strictly ordered preference list. This is obviously often unrealistic in practice, and several relaxations have been proposed, including the following two common ones: one is to allow an incomplete list, i.e., a man is permitted to accept only a subset of the women and vice versa. The other is to allow a preference list including ties. Fortunately, it is known that both relaxed problems can still be solved in polynomial time. In this paper, we show that the situation changes substantially if we allow both relaxations (incomplete lists and ties) at the same time: the problem not only becomes NP-hard, but also the optimal cost version has no approximation algorithm achieving the approximation ratio of N1-Ɛ, where N is the instance size, unless P=NP. We consider the variant of the classical Stable Marriage problem where preference lists can be incomplete and may contain ties. In such a setting, finding a stable matching of maximum size is NP-hard. We study the parameterized complexity of this problem, where the parameter can be the number of ties, the maximum or the overall length of ties. We also investigate the applicability of a local search algorithm for the problem. Finally, we examine the possibilities for giving an FPT algorithm or an FPT approximation algorithm for finding an egalitarian or a minimum regret matching. The Stable Marriage Problem and its many variants have been widely studied in the literature (Gusfield and Irving, The Stable Marriage Problem: Structure and Algorithms, MIT Press, Cambridge, MA, 1989; Roth and Sotomayor, Two-sided matching: a study in game-theoretic modeling and analysis, Econometric Society Monographs, vol. 18, Cambridge University Press, Cambridge, 1990; Knuth, Stable Marriage and its Relation to Other Combinatorial Problems, CRM Proceedings and Lecture Notes, vol. 10, American Mathematical Society, Providence, RI, 1997), partly because of the inherent appeal of the problem, partly because of the elegance of the associated structures and algorithms, and partly because of important practical applications, such as the National Resident Matching Program (Roth, J. Political Economy 92(6) (1984) 991) and similar large-scale matching schemes. Here, we present the first comprehensive study of variants of the problem in which the preference lists of the participants are not necessarily complete and not necessarily totally ordered. We show that, under surprisingly restrictive assumptions, a number of these variants are hard, and hard to approximate. The key observation is that, in contrast to the case where preference lists are complete or strictly ordered (or both), a given problem instance may admit stable matchings of different sizes. In this setting, examples of problems that are hard are: finding a stable matching of maximum or minimum size, determining whether a given pair is stableeven if the indifference takes the form of ties on one side only, the ties are at the tails of lists, there is at most one tie per list, and each tie is of length 2; and finding, or approximating, both an egalitarian' and a minimum regret' stable matching. However, we give a 2-approximation algorithm for the problems of finding a stable matching of maximum or minimum size. We also discuss the significant implications of our results for practical matching schemes. We present the first complete algorithm for the SMTI problem, the stable marriage problem with ties and incomplete lists. We do this in the form of a constraint programming encoding of the problem. With this we are able to carry out the first empirical study of the complete solution of SMTI instances. In the stable marriage problem (SM) [5] we have n men and n women. Each man ranks the women, giving himself a preference list. Similarly each woman ranks the men, giving herself a preference list. The problem is then to marry men and women such that they are stable i.e. such that there is no incentive for individuals to divorce and elope. This problem is polynomial time solvable. However, when preference lists contain ties and are incomplete (SMTI) the problem of determining if there is a stable matching of size n is then NP-complete, as is the optimisation problem of finding the largest or smallest stable matching [6, 10]. In this paper we present constraint programming solutions for the SMTI decision and optimisation problems, a problem generator for random instances of SMTI, and an empirical study of this problem. 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 @math -approximation algorithm for the general version of this problem, improving upon the recent @math -approximation algorithm of Kiraly. Interest in such algorithms arises because of the problem's application to centralized matching schemes, the best known of which involve the assignment of graduating medical students to hospitals in various countries. 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. 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. The stable marriage problem has recently been studied in its general setting, where both ties and incomplete lists are allowed. It is NP-hard to find a stable matching of maximum size, while any stable matching is a maximal matching and thus trivially we can obtain a 2-approximation algorithm. In this article, we give the first nontrivial result for approximation of factor less than two. Our algorithm achieves an approximation ratio of 2 (1 p L−2) for instances in which only men have ties of length at most L. When both men and women are allowed to have ties but the lengths are limited to two, then we show a ratio of 13 7( 1.1052). In an instance G = (A union B, E) of the stable marriage problem with strict and possibly incomplete preference lists, a matching M is popular if there is no matching M0 where the vertices that prefer M' to M outnumber those that prefer M to M'. All stable matchings are popular and there is a simple linear time algorithm to compute a maximum-size popular matching. More generally, what we seek is a min-cost popular matching where we assume there is a cost function c : E -> Q. However there is no polynomial time algorithm currently known for solving this problem. Here we consider the following generalization of a popular matching called a popular half-integral matching: this is a fractional matching x = (M_1 + M_2) 2, where M1 and M2 are the 0-1 edge incidence vectors of matchings in G, such that x satisfies popularity constraints. We show that every popular half-integral matching is equivalent to a stable matching in a larger graph G^*. This allows us to solve the min-cost popular half-integral matching problem in polynomial time. The stable marriage (SM) problem has a wide variety of practical applications, ranging from matching resident doctors to hospitals, to matching students to schools or, more generally, to any two-sided market. In the classical formulation, n men and n women express their preferences (via a strict total order) over the members of the other sex. Solving an SM problem means finding a stable marriage where stability is an envy-free notion: no man and woman who are not married to each other would both prefer each other to their partners or to being single. We consider both the classical stable marriage problem and one of its useful variations (denoted SMTI (Stable Marriage with Ties and Incomplete lists)) where the men and women express their preferences in the form of an incomplete preference list with ties over a subset of the members of the other sex. Matchings are permitted only with people who appear in these preference lists, and we try to find a stable matching that marries as many people as possible. Whilst the SM problem is polynomial to solve, the SMTI problem is NP-hard. We propose to tackle both problems via a local search approach, which exploits properties of the problems to reduce the size of the neighborhood and to make local moves efficiently. We empirically evaluate our algorithm for SM problems by measuring its runtime behavior and its ability to sample the lattice of all possible stable marriages. We evaluate our algorithm for SMTI problems in terms of both its runtime behavior and its ability to find a maximum cardinality stable marriage. Experimental results suggest that for SM problems, the number of steps of our algorithm grows only as O(n log(n)), and that it samples very well the set of all stable marriages. It is thus a fair and efficient approach to generate stable marriages. Furthermore, our approach for SMTI problems is able to solve large problems, quickly returning stable matchings of large and often optimal size, despite the NP-hardness of this problem. We investigate the complexity of approximately counting stable roommate assignments in two models: (i) the k-attribute model, in which the preference lists are determined by dot products of ''preference vectors'' with ''attribute vectors'' and (ii) the k-Euclidean model, in which the preference lists are determined by the closeness of the ''positions'' of the people to their ''preferred positions''. Exactly counting the number of assignments is #P-complete, since Irving and Leather demonstrated #P-completeness for the special case of the stable marriage problem (Irving and Leather, 1986 [11]). We show that counting the number of stable roommate assignments in the k-attribute model (#k-attribute SR, k>=4) and the 3-Euclidean model (#k-Euclidean SR, k>=3) is interreducible, in an approximation-preserving sense, with counting independent sets (of all sizes) (#IS) in a graph, or counting the number of satisfying assignments of a Boolean formula (#SAT). This means that there can be no FPRAS for any of these problems unless NP=RP. As a consequence, we infer that there is no FPRAS for counting stable roommate assignments (#SR) unless NP=RP. Utilizing previous results by Chebolu, Goldberg and Martin (2010) [3], we give an approximation-preserving reduction from counting the number of independent sets in a bipartite graph (#BIS) to counting the number of stable roommate assignments both in the 3-attribute model and in the 2-Euclidean model. #BIS is complete with respect to approximation-preserving reductions in the logically-defined complexity class #RH@P"1. Hence, our result shows that an FPRAS for counting stable roommate assignments in the 3-attribute model would give an FPRAS for all #RH@P"1. We also show that the 1-attribute stable roommate problem always has either one or two stable roommate assignments, so the number of assignments can be determined exactly in polynomial time. We introduce a restriction of the stable roommates problem in which roommate pairs are ranked globally. In contrast to the unrestricted problem, weakly stable matchings are guaranteed to exist, and additionally, they can be found in polynomial time. However, it is still the case that strongly stable matchings may not exist, and so we consider the complexity of finding weakly stable matchings with various desirable properties. In particular, we present a polynomial-time algorithm to find a rank-maximal (weakly stable) matching. This is the first generalization of an algorithm due to [ 06] to a nonbipartite setting. Also, we describe several hardness results in an even more restricted setting for each of the problems of finding weakly stable matchings that are of maximum size, are egalitarian, have minimum regret, and admit the minimum number of weakly blocking pairs. 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 n^1^-^@e, for any @e>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 @d, for some @d>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. The stable roommates problem is a well-known problem of matching n people into n2 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(n2) 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(n2) algorithm that finds one stable partition which in turn gives all the odd parties. We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose–accept rounds executed by the Gale–Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about the problem instance; for each participant, knowing only its local neighbourhood is enough. In distributed-systems parlance, this means that if each person has only a constant number of acceptable partners, an almost stable matching emerges after a constant number of synchronous communication rounds. We apply our results to give a distributed (2+e)-approximation algorithm for maximum-weight matching in bicoloured graphs and a centralised randomised constant-time approximation scheme for estimating the size of a stable matching. 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. 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 n2 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. Stable Marriage is a fundamental problem to both computer science and economics. Four well-known NP-hard optimization versions of this problem are the Sex-Equal Stable Marriage (SESM), Balanced Stable Marriage (BSM), max-Stable Marriage with Ties (max-SMT) and min-Stable Marriage with Ties (min-SMT) problems. In this paper, we analyze these problems from the viewpoint of Parameterized Complexity. We conduct the first study of these problems with respect to the parameter treewidth. First, we study the treewidth @math of the primal graph. We establish that all four problems are W[1]-hard. In particular, while it is easy to show that all four problems admit algorithms that run in time @math , we prove that all of these algorithms are likely to be essentially optimal. Next, we study the treewidth @math of the rotation digraph. In this context, the max-SMT and min-SMT are not defined. For both SESM and BSM, we design (non-trivial) algorithms that run in time @math . Then, for both SESM and BSM, we also prove that unless SETH is false, algorithms that run in time @math do not exist for any fixed @math . We thus present a comprehensive, complete picture of the behavior of central optimization versions of Stable Marriage with respect to treewidth.