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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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 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 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. 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. 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 n1e. 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.