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The stable matching problem (also known as the stable marriage problem) is a well-known problem of matching men to women, so that no man and woman, who are not married to each other, both prefer each other. Such a 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 stable marriage problem, both men and women express a strict preference order over the members of the other sex, in a qualitative way. Here, we consider stable marriage problems with weighted preferences: each man (resp., woman) provides a score for each woman (resp., man). Such problems are more expressive than the classical stable marriage problems. Moreover, in some real-life situations, it is more natural to express scores (to model, for example, profits or costs) rather than a qualitative preference ordering. In this context, we define new notions of stability and optimality, and we provide algorithms to find marriages that are stable and/or optimal according to these notions. While expressivity greatly increases by adopting weighted preferences, we show that, in most cases, the desired solutions can be found by adapting existing algorithms for the classical stable marriage problem. We also consider the manipulability properties of the procedures that return such stable marriages. While we know that all procedures are manipulable by modifying the preference lists or by truncating them, here, we consider if manipulation can occur also by just modifying the weights while preserving the ordering and avoiding truncation. It turns out that, by adding weights, in some cases, we may increase the possibility of manipulating, and this cannot be avoided by any reasonable restriction on the weights. Full article

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. Full article

In the stable marriage problem, any instance admits the so-called man-optimal stable matching, in which every man is assigned the best possible partner. However, there are instances for which all men receive low-ranked partners even in the man-optimal stable matching. In this paper we consider the problem of improving the man-optimal stable matching by changing only one man’s preference list. We show that the optimization variant and the decision variant of this problem can be solved in time O(n³) and O(n²), respectively, where n is the number of men (women) in an input. We further extend the problem so that we are allowed to change k men’s preference lists. We show that the problem is W[1]-hard with respect to the parameter k and give O(n^2k+1)-time and O(n^k+1)-time exact algorithms for the optimization and decision variants, respectively. Finally, we show that the problems become easy when k = n; we give O(n^2.5 log n)-time and O(n²)-time algorithms for the optimization and decision variants, respectively. Full article