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

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

Solving Stable Matching Problems via Cooperative Parallel Local Search

International audienceStable matching problems and its variants have several practical applications, like the Hospital/Residents problem, stable roommates problem or bipartite market sharing. An important generalization problem is the SMTI which allows for incompleteness and ties in the user's preference lists. Finding a maximal size stable matching for SMTI is compu-tationally difficult. We developed a Local Search method to solve SMTI using the Adaptive Search algorithm and present experimental evidence that this approach is much more efficient than state-of-the-art exact and approximate methods (in terms of both computational effort required and quality of solution). We also tried a parallel version of our algorithm. For this we reused the Cooperative Parallel Local Search framework (CPLS) we designed. CPLS is a highly parametric framework for the execution in parallel of local search solvers allowing them to cooperate though communication. The cooperative parallel version of our local search algorithm improves performance so much that very large and hard instances can be solved quickly

Reasoning with incomplete and imprecise preferences

Preferences are present in many real life situations but it is often difficult to quantify them giving a precise value. Sometimes preference values may be missing because of privacy reasons or because they are expensive to obtain or to produce. In some other situations the user of an automated system may have a vague idea of whats he wants. In this thesis we considered the general formalism of soft constraints, where preferences play a crucial role and we extended such a framework to handle both incomplete and imprecise preferences. In particular we provided new theoretical frameworks to handle such kinds of preferences. By admitting missing or imprecise preferences, solving a soft constraint problem becomes a different task. In fact, the new goal is to find solutions which are the best ones independently of the precise value the each preference may have. With this in mind we defined two notions of optimality: the possibly optimal solutions and the necessary optimal solutions, which are optimal no matter we assign a precise value to a missing or imprecise preference. We provided several algorithms, bases on both systematic and local search approaches, to find such kind of solutions. Moreover, we also studied the impact of our techniques also in a specific class of problems (the stable marriage problems) where imprecision and incompleteness have a specific meaning and up to now have been tackled with different techniques. In the context of the classical stable marriage problem we developed a fair method to randomly generate stable marriages of a given problem instance. Furthermore, we adapted our techniques to solve stable marriage problems with ties and incomplete lists, which are known to be NP-hard, obtaining good results both in terms of size of the returned marriage and in terms of steps need to find a solution

Unveiling Hidden Values of Optimization Models with Metaheuristic Approach

Considering that the decision making process for constrained optimization problem is based on modeling, there is always room for alternative solutions because there is usually a gap between the model and the real problem it depicts. This study looks into the problem of finding such alternative solutions, the non-optimal solutions of interest for constrained optimization models, the SoI problem. SoI problems subsume finding feasible solutions of interest (FoIs) and infeasible solutions of interest (IoIs). In all cases, the interest addressed is post-solution analysis in one form or another. Post-solution analysis of a constrained optimization model occurs after the model has been solved and a good or optimal solution for it has been found. At this point, sensitivity analysis and other questions of import for decision making come into play and for this purpose the SoIs can be very valuable. An evolutionary computation approach (in particular, a population-based metaheuristic) is proposed for solving the SoI problem and a systematic approach with a feasible-infeasible- two-population genetic algorithm is demonstrated. In this study, the effectiveness of the proposed approach on finding SoIs is demonstrated with generalized assignment problems and generalized quadratic assignment problems. Also, the applications of the proposed approach on the multi-objective optimization and robust-optimization issues are examined and illustrated with two-sided matching problems and flowshop scheduling problems respectively

The Strength of Absent Ties: Social Integration via Online Dating

We used to marry people to which we were somehow connected to: friends of friends, schoolmates, neighbours. Since we were more connected to people similar to us, we were likely to marry someone from our own race. However, online dating has changed this pattern: people who meet online tend to be complete strangers. Given that one-third of modern marriages start online, we investigate theoretically, using random graphs and matching theory, the effects of those previously absent ties in the diversity of modern societies. We find that when a society benefits from previously absent ties, social integration occurs rapidly, even if the number of partners met online is small. Our findings are consistent with the sharp increase in interracial marriages in the U.S. in the last two decades

A systematic reflection upon dual career couples

Particularly among the highly educated, a persistent upward trend in female employment rates has characterized western industrialized countries in the last decades. Yet, strong gender inequalities persist in the career chances of equally highly qualified men and women. Women are still underrepresented in executive/ leading positions in both the private and public sector of the economy. We argue that such gender inequalities are also due to the fact that the majority of highly educated women lives with an equally highly educated partner. For these women the realization of dual careers becomes ever more important and represents an essential prerequisite for their own professional development. Following Phyllis Moen's 'linked lives' idea, we will argue that the achievement or failure of dual-career arrangements is a 'social-relational process' (Moen 2003a: 10) and that partners' lives are embedded with and influenced by each other. In particular, we will discuss how this entwining occurs, which processes at different levels play a role, and how these different processes interact with each other. Finally, we will give some suggestions on the direction for future research. -- Vor allem bei Hochqualifizierten charakterisiert ein andauernder Aufwärtstrend weiblicher Beschäftigungsraten die westlichen Industrieländer in den letzten Jahrzehnten. Jedoch herrscht bei den Karrierechancen von gleich qualifizierten Männern und Frauen noch immer eine große Geschlechterungleichheit vor. Frauen sind in führenden/leitenden Positionen sowohl im privatwirtschaftlichen als auch im öffentlichen Beschäftigungssektor noch immer unterrepräsentiert. Wir argumentieren, dass solche Geschlechterungleichheiten auf den Umstand zurückzuführen sind, dass die Mehrzahl hoch gebildeter Frauen mit gleichermaßen hoch gebildeten Partnern zusammenlebt. Für diese Frauen gewinnt die Realisierung von Doppelkarrieren an Bedeutung und stellt eine entscheidende Voraussetzung für ihre eigene professionelle Entwicklung dar. In Anlehnung an Phyllis Moens "linked lives" Idee werden wir diskutieren, dass der Erfolg oder das Scheitern von Doppelkarrierenarrangements ein "social- relational process" (Moen 2003a: 10) ist und dass die Lebensverläufe der Partner miteinander verwoben und voneinander abhängig sind. Im Einzelnen werden wir diskutieren, wie diese Verflechtung sich gestaltet, welche Prozesse auf verschiedenen Ebenen eine Rolle spielen und wie diese Prozesse miteinander interagieren. Zum Schluss werden wir Vorschläge für die Richtung zukünftiger Forschungen formulieren.

Partner Choices in Long Established Migrant Communities in Belgium

This paper aims to shed light on the partner choices of Moroccan, Turkish, Congolese, and Algerian migrants in Belgium. Three partner choices are distinguished: marrying a partner from the country of origin (partner migration), marrying a local co-ethnic partner, and establishing a mixed marriage. We focused on the role of migration history and transnational links, culture (religion, language), skin colour and structural characteristics of the district migrants live in (mainly community size) to gain further insight into the partner choices of migrants in Belgium. Our data comprise an extraction of the Belgian national register (2001-2008) and focus on first marriages among first, 1.5, and second generation migrants of Moroccan, Turkish, Algerian, and Congolese origin (N=52,142). We apply a multinomial logistic multilevel design to simultaneously incorporate individual and contextual effects at the district level. The main conclusion from this paper is that the partner selection pattern in early 21st century Belgian society still bears the traces of the starting conditions that migrant groups experienced when they first entered the country. While this continuity is important to understand the situation citizens with a migrant origin have to deal with today, it does not make change impossible. In fact, for the Turkish and Moroccan group, research recently showed a quite strong decline in transnational marriages and a modest increase in mixed marriages. These are indications that after 50 years of migration a transition towards full inclusion in Belgian society is not beyond reach. The conditions analysed in this paper, namely the strength of transnational networks, the cultural boundaries and the ethnic community size, may help to understand why this inclusion takes such a long period of time

Workforce scheduling and planning : a combinatorial approach

This thesis investigates solution methodologies for concrete combinatorial problems in scheduling and planning. In all considered problems, it is assumed that the available information does not change over time; hence these problems have a deterministic structure. The problems studied in this thesis are divided into two groups; \workforce scheduling" and \planning". In workforce scheduling, the center problem is to build a schedule of tasks and technicians. It is assumed that the time line is split into workdays. In every workday, tasks must be grouped as sequences, each being performed by a team of technicians. Skill requirements of every task in a sequence must be met by the assigned team. This scheduling problem with some other aspects is di??cult to solve quickly and e??ciently. We developed a Mixed Integer Programming (MIP) based heuristic approach to tackle this complex scheduling problem. Our MIP model is basically a formulation of the matching problem on bipartite graphs and it enabled us to have a global way of assigning technicians to tasks. It is capable of revising technician-task allocations and performs very well, especially in the case of rare skills. A workday schedule of the aforementioned problem includes many-to-one type workforce assignments. As the second problem in workforce scheduling, stability of these workforce assignments is investigated. The stability de??nition of Gale-Shapley on the Marriage model is extended to the setting of multi-skill workforce assignments. It is shown that ??nding stable assignments is NP-hard. In some special cases stable assignments can be constructed in polynomial time. For the general case, we give linear inequalities of binary variables that describe the set of stable assignments. We propose a MIP model including these linear inequalities. To the best of our knowledge, the Gale-Shapley stability is not studied under the multi-skill workforce scheduling framework so far in the literature. The closed form description of stable assignments is also the ??rst embedding of the Gale-Shapley stability concept into an NP-complete problem. In the second problem group, two vehicle related problems are studied; the "dial a ride problem" and the "vehicle refueling problem". In the former, the goal is to check whether a list of pick-up and delivery tasks can be achieved under several timing constraints. It is shown this feasibility testing can be done in linear time using interval graphs. In the vehicle refueling problem, the goal is to make refueling decisions to reach a destination such that the cost of the travel is minimized. A greedy algorithm is presented to ??nd optimal refueling decisions. Moreover, it is shown that the vehicle refueling problem is equivalent to a ow problem on a special network

Stable Matchings with Restricted Preferences: Structure and Complexity

In the stable marriage (SM) problem, there are two sets of agents–traditionally referred to as men and women–and each agent has a preference list that ranks (a subset of) agents of the opposite sex. The goal is to find a matching between men and women that is stable in the sense that no man-woman pair mutually prefer each other to their assigned partners. In a seminal work, Gale and Shapley showed that stable matchings always exist, and described an efficient algorithm for finding one. Irving and Leather defined the rotation poset of an SM instance and showed that it determines the structure of the set of stable matchings of the instance. They further showed that every finite poset can be realized as the rotation poset of some SM instance. Consequently, many problems–such as counting stable matchings and finding certain “fair” stable matchings–are computationally intractable (NP-hard) in general. In this paper, we consider SM instances in which certain restrictions are placed on the preference lists. We show that three natural preference models?k-bounded, k-attribute, and (k1, k2)-list–can realize arbitrary rotation posets for constant values of k. Hence even in these highly restricted preference models, many stable matching problems remain intractable. In contrast, we show that for any fixed constant k, the rotation posets of k-range instances are highly restricted. As a consequence, we show that exactly counting and uniformly sampling stable matchings, finding median, sex-equal, and balanced stable matchings are fixed-parameter tractable when parameterized by the range of the instance. Thus, these problems can be solved in polynomial time on instances of the k-range model for any fixed constant k