We present new integer linear programming (ILP) models for NP-hard optimisation problems in instances of the Stable Marriage problem with Ties and Incomplete lists (SMTI) and its many-to-one generalisation, the Hospitals Residents problem with Ties (HRT). These models can be used to efficiently solve these optimisation problems when applied to (i) instances derived from real-world applications, and (ii) larger instances that are randomly-generated. In the case of SMTI, we consider instances arising from the pairing of children with adoptive families, where preferences are obtained from a quality measure of each possible pairing of child to family. In this case we seek a maximum weight stable matching. We present new algorithms for preprocessing instances of SMTI with ties on both sides, as well as new ILP models. Algorithms based on existing state-of-the-art models only solve 6 of our 22 real-world instances within an hour per instance, and our new models solve all 22 instances within a mean runtime of 60 seconds. For HRT, we consider instances derived from the problem of assigning junior doctors to foundation posts in Scottish hospitals. Here we seek a maximum size stable matching. We show how to extend our models for SMTI to the HRT case. For the real instances, we reduce the mean runtime from an average of 144 seconds when using state-of-the-art methods, to 3 seconds when using our new ILP-based algorithms. We also show that our models outperform considerably state-of-the-art models on larger randomly-generated instances of SMTI and HRT.
The Stable Marriage problem with Ties, Incomplete lists and Symmetric preferences (SMTI-SYM) is a restriction of SMTI-GRP such that (i) for each man--woman pair @math , the rank of @math in @math 's list, i.e., the integer @math such that @math belongs to the @math th tie in @math 's list, is equal to the rank of @math in @math 's list, and (ii) the weight of @math is precisely this integer @math . Finding a maximum size stable matching in an instance of SMTI-SYM is @math -hard, and therefore the same result holds for SMTI-GRP @cite_7 . Given an instance of SMTI-GRP, if the goal is to find a matching that maximises the total weight rather than the total size, this problem is @math -hard also @cite_19 .
@cite_19 @cite_7
178770064 2047128428 2736636075 1996180590
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. 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. 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. 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).