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

Guest editorial: Special issue on matching under preferences

No abstract available

Task Assignment with Autonomous and Controlled Agents

We analyse assignment problems in which not all agents are controlled by the central planner. The autonomous agents search for vacant tasks guided by their own preference orders defined over subsets of the available tasks. The goal of the central planner is to maximise the total value of the assignment, taking into account the behaviour of the uncontrolled agents. This setting can be found in numerous real-world situations, ranging from organisational economics to "crowdsourcing" and disaster response. We introduce the Disjunctively Constrained Knapsack Game and show that its unique Nash equilibrium reveals the optimal assignment for the controlled agents. This result allows us to find the solution of the problem using mathematical programming techniques.

Editorial: special issue on matching under preferences

This special issue of Algorithms is devoted to the study of matching problems involving ordinal preferences from the standpoint of algorithms and complexit

The marriage problem: from the bar of appointments to the agency

We study the stable marriage problem from different points of view. We proposed a microscopic dynamic that lead the system to a stationary state that we are able to characterize analytically. Then, we derive a thermodynamical description of the Nash equilibrium states of the system that agree very well with the results of Monte Carlo simulations. Finally, through large scale numerical simulations we compare the Global Optimum of the society with the stable marriage of lower energy. We showed that both states are strongly correlated and that the selffish attitude results in a benefit for most of the practitioners belonging to blocking pairs in the Global Optimum of the society.Comment: 15 pages, 9 figures. To be published in Physica A (2005