Local search for stable marriage problems with ties and incomplete lists

The stable marriage 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. We consider a useful variation of the stable marriage problem, where the men and women express their preferences using a 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. In this setting, we study the problem of finding a stable matching that marries as many people as possible. 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. This problem is NP-hard. We tackle this problem using local search, exploiting properties of the problem to reduce the size of the neighborhood and to make local moves efficiently. Experimental results show that this approach is able to solve large problems, quickly returning stable matchings of large and often optimal size.Comment: 12 pages, Proc. PRICAI 2010 (11th Pacific Rim International Conference on Artificial Intelligence), Byoung-Tak Zhang and Mehmet A. Orgun eds., Springer LNA

Parameterized Complexity of Stable Roommates with Ties and Incomplete Lists Through the Lens of Graph Parameters

We continue and extend previous work on the parameterized complexity analysis of the NP-hard Stable Roommates with Ties and Incomplete Lists problem, thereby strengthening earlier results both on the side of parameterized hardness as well as on the side of fixed-parameter tractability. Other than for its famous sister problem Stable Marriage which focuses on a bipartite scenario, Stable Roommates with Incomplete Lists allows for arbitrary acceptability graphs whose edges specify the possible matchings of each two agents (agents are represented by graph vertices). Herein, incomplete lists and ties reflect the fact that in realistic application scenarios the agents cannot bring all other agents into a linear order. Among our main contributions is to show that it is W[1]-hard to compute a maximum-cardinality stable matching for acceptability graphs of bounded treedepth, bounded tree-cut width, and bounded feedback vertex number (these are each time the respective parameters). However, if we "only" ask for perfect stable matchings or the mere existence of a stable matching, then we obtain fixed-parameter tractability with respect to tree-cut width but not with respect to treedepth. On the positive side, we also provide fixed-parameter tractability results for the parameter feedback edge set number

Size versus stability in the marriage problem

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 n1−ε, for any ε>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 δ, for some δ>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

Improving solution times for stable matching problems through preprocessing

We present new theory, heuristics, and algorithms for preprocessing instances of the Stable Marriage problem with Ties and Incomplete lists (SMTI) and the Hospitals/Residents problem with Ties (HRT). Instances of these problems can be preprocessed by removing from the preference lists of some agents entries such that the set of stable matchings is not affected. Removing such entries reduces the problem size, creating smaller models that can be more easily solved by integer programming (IP) solvers. The new theorems are the first to describe when preference list entries can be removed from instances of HRT when ties are present on both sides, and also extend existing results on preprocessing instances of SMTI. A number of heuristics, as well as an IP model and a graph-based algorithm, are presented to find and perform this preprocessing. Experimental results show that our new graph-based algorithm achieves a 44% reduction in the average running time to find a maximum weight stable matching in real-world instances of SMTI compared to existing preprocessing techniques, and 80% compared to not using preprocessing. We also show that, when solving MAX-HRT instances with ties on both sides, our new techniques can reduce runtimes by up to 55%

Algorithmic aspects of stable matching problems

The Stable Marriage problem (SM), the Hospitals/Residents problem (HR) and the Stable Roommates problem (SR) are three classical stable matching problems that were first studied by Gale and Shapley in 1962. These problems have widespread practical application in centralised automated matching schemes, which assign applicants to posts based on preference lists and capacity constraints in both the UK and internationally. Within such schemes it is often the case that an agent's preference list may be incomplete, and agents may also be allowed to express indifference in the form of ties. In the presence of ties, three stability criteria can be defined, namely weak stability, strong stability and super-stability. In this thesis we consider stable matching problems from an algorithmic point of view. Some of the problems that we consider are derived from new stable matching models, whilst others are obtained from existing stable matching models involving ties and incomplete lists, with additional natural restrictions on the problem instance. Furthermore, we also explore the use of constraint programming with both SM and HR. We first study a new variant of the Student-Project Allocation problem in which each student ranks a set of acceptable projects in preference order and similarly each lecturer ranks his available projects in preference order. In this context, two stability definitions can be identified, namely weak stability and strong stability. We show that the problem of finding a maximum weakly stable matching is NP-hard. However, we describe two 2-approximation algorithms for this problem. Regarding strong stability, we describe a polynomial-time algorithm for finding such a matching or reporting that none exists. Next we investigate SM with ties and incomplete lists (SMTI), and HR with ties (HRT), where the length of each agent's list is subject to an upper bound. We present both polynomial-time algorithms and NP-hardness results for a range of problems that are derived from imposing upper bounds on the length of the lists on one or both sides. We also consider HRT, and SR with ties and incomplete lists (SRTI), where the preference lists of one or both sets of agents (as applicable) are derived from one or two master lists in which agents are ranked. For super-stability, in the case of each of HRT and SRTI with a master list, we describe a linear-time algorithm that simplifies the algorithm used in the general case. In the case of strong stability, for each of HRT and SRTI with a master list, we describe an algorithm that is faster than that for the general case. We also show that, given an instance I of SRTI with a master list, the problem of finding a weakly stable matching is polynomial-time solvable. However, we show that given such an I, the problem of finding a maximum weakly stable matching is NP-hard. Other new stable matching models that we study are the variants of SMTI and SRTI with symmetric preferences. In this context we consider two models that are derived from alternative ways of interpreting the rank of an agent in the presence of ties. For both models we show that deciding if a complete weakly stable matching exists is NP-complete. Then for one of the models we show that each of the problem of finding a minimum regret and an egalitarian weakly stable matching is NP-hard and that the problem of determining if a (man,woman) pair belongs to a weakly stable matching is NP-complete. We then describe algorithms for each of the problems of finding a super-stable matching and a strongly stable matching, or reporting that none exists, given instances of SRTI and HRT with symmetric preferences (regardless of how the ranks are interpreted). Finally, we use constraint programming techniques to model instances of SM and HR. We describe two encodings of SM in terms of a constraint satisfaction problem. The first model for SM is then extended to the case of HR. This encoding for HR is then extended to create a model for HRT under weak stability. Using this encoding we can obtain, with the aid of search, all the weakly stable matchings, given an instance of HRT

Hardness of Instance Generation with Optimal Solutions for the Stable Marriage Problem

In a variant of the stable marriage problem where ties and incomplete lists are allowed, finding a stable matching of maximum cardinality is known to be NP-hard. There are a lot of experimental studies for evaluating the performance of approximation algorithms or heuristics, using randomly generated or artificial instances. One of standard evaluation methods is to compare an algorithm's solution with an optimal solution, but finding an optimal solution itself is already hard. In this paper, we investigate the possibility of generating instances with known optimal solutions. We propose three instance generators based on a known random generation algorithm, but unfortunately show that none of them meet our requirements, implying a difficulty of instance generation in this approach

Local search for stable marriage problems

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 a 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) 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 lists, an 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 evaluate empirically our algorithm for SM problems by measuring its runtime behaviour 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 behaviour and its ability to find a maximum cardinality stable marriage.For SM problems, the number of steps of our algorithm grows only as O(nlog(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.Comment: 12 pages, Proc. COMSOC 2010 (Third International Workshop on Computational Social Choice

Solving Hard Stable Matching Problems Involving Groups of Similar Agents

Many important stable matching problems are known to be NP-hard, even when strong restrictions are placed on the input. In this paper we seek to identify structural properties of instances of stable matching problems which will allow us to design efficient algorithms using elementary techniques. We focus on the setting in which all agents involved in some matching problem can be partitioned into k different types, where the type of an agent determines his or her preferences, and agents have preferences over types (which may be refined by more detailed preferences within a single type). This situation would arise in practice if agents form preferences solely based on some small collection of agents' attributes. We also consider a generalisation in which each agent may consider some small collection of other agents to be exceptional, and rank these in a way that is not consistent with their types; this could happen in practice if agents have prior contact with a small number of candidates. We show that (for the case without exceptions), several well-studied NP-hard stable matching problems including Max SMTI (that of finding the maximum cardinality stable matching in an instance of stable marriage with ties and incomplete lists) belong to the parameterised complexity class FPT when parameterised by the number of different types of agents needed to describe the instance. For Max SMTI this tractability result can be extended to the setting in which each agent promotes at most one `exceptional' candidate to the top of his/her list (when preferences within types are not refined), but the problem remains NP-hard if preference lists can contain two or more exceptions and the exceptional candidates can be placed anywhere in the preference lists, even if the number of types is bounded by a constant.Comment: Results on SMTI appear in proceedings of WINE 2018; Section 6 contains work in progres

Maximum Cardinality Popular Matchings in Strict Two-sided Preference Lists

We consider the problem of computing a maximum cardinality {\em popular} matching in a bipartite graph G = (\A\cup\B, E) where each vertex u \in \A\cup\B ranks its neighbors in a strict order of preference. This is the same as an instance of the {\em stable marriage} problem with incomplete lists. A matching $M^*$ is said to be popular if there is no matching $M$ such that more vertices are better off in $M$ than in $M^*$. \smallskip Popular matchings have been extensively studied in the case of one-sided preference lists, i.e., only vertices of \A have preferences over their neighbors while vertices in \B have no preferences; polynomial time algorithms have been shown here to determine if a given instance admits a popular matching or not and if so, to compute one with maximum cardinality. It has very recently been shown that for two-sided preference lists, the problem of determining if a given instance admits a popular matching or not is NP-complete. However this hardness result assumes that preference lists have {\em ties}. When preference lists are {\em strict}, it is easy to show that popular matchings always exist since stable matchings always exist and they are popular. But the complexity of computing a maximum cardinality popular matching was unknown. In this paper we show an $O(mn)$ algorithm for this problem, where n = |\A| + |\B| and $m = |E|$