The joy of matching

Here, the authors discuss matching problems and how the Gale-Shapley algorithm solves them, while also explaining some matching techniques

Profile-Based Optimal Matchings in the Student-Project Allocation Problem

In the Student/Project Allocation problem (spa) we seek to assign students to individual or group projects offered by lecturers. Students provide a list of projects they find acceptable in order of preference. Each student can be assigned to at most one project and there are constraints on the maximum number of students that can be assigned to each project and lecturer. We seek matchings of students to projects that are optimal with respect to profile, which is a vector whose rth component indicates how many students have their rth-choice project. We present an efficient algorithm for finding agreedy maximum matching in the spa context – this is a maximum matching whose profile is lexicographically maximum. We then show how to adapt this algorithm to find a generous maximum matching – this is a matching whose reverse profile is lexicographically minimum. Our algorithms involve finding optimal flows in networks. We demonstrate how this approach can allow for additional constraints, such as lecturer lower quotas, to be handled flexibly

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

Stable marriage with general preferences

We propose a generalization of the classical stable marriage problem. In our model, the preferences on one side of the partition are given in terms of arbitrary binary relations, which need not be transitive nor acyclic. This generalization is practically well-motivated, and as we show, encompasses the well studied hard variant of stable marriage where preferences are allowed to have ties and to be incomplete. As a result, we prove that deciding the existence of a stable matching in our model is NP-complete. Complementing this negative result we present a polynomial-time algorithm for the above decision problem in a significant class of instances where the preferences are asymmetric. We also present a linear programming formulation whose feasibility fully characterizes the existence of stable matchings in this special case. Finally, we use our model to study a long standing open problem regarding the existence of cyclic 3D stable matchings. In particular, we prove that the problem of deciding whether a fixed 2D perfect matching can be extended to a 3D stable matching is NP-complete, showing this way that a natural attempt to resolve the existence (or not) of 3D stable matchings is bound to fail.Comment: This is an extended version of a paper to appear at the The 7th International Symposium on Algorithmic Game Theory (SAGT 2014

Finding “diversity levers” in the core library and information science curriculum: a social justice imperative

In this exploratory study, the researchers examined the core library and information science (LIS) curriculum, looking for diversity levers, or conceptual access points, where transformative academic knowledge related to diversity and social justice could be meaningfully integrated. Multicultural curriculum reform, conceptualized as a social justice approach, was the guiding framework for the research design and analysis. The researchers began by establishing what constitutes the core curriculum and essential knowledge taught across thirty-six ALA-accredited master’s of library and information science degree programs. These data were then used to construct a survey that went to one hundred LIS faculty at ALA institutions who provided pedagogical knowledge, ideas, and resources for infusing diversity and social justice into the core curriculum. The findings suggest that there are certain core LIS courses that have explicit diversity levers, or areas where there are natural connections to diversity and social justice content, while others have emergent or implicit diversity levers. The differences among these types of diversity levers are explained, and some of the pedagogical resources that were shared by the survey respondents are included. The Information Technology core course shows the most promise for integrating diversity and social justice pedagogies

The Stable Roommates problem with short lists

We consider two variants of the classical Stable Roommates problem with Incomplete (but strictly ordered) preference lists SRI that are degree constrained, i.e., preference lists are of bounded length. The first variant, EGAL d-SRI, involves finding an egalitarian stable matching in solvable instances of SRI with preference lists of length at most d. We show that this problem is NP-hard even if d=3. On the positive side we give a (2d+3)/7-approximation algorithm for d={3,4,5} which improves on the known bound of 2 for the unbounded preference list case. In the second variant of SRI, called d-SRTI, preference lists can include ties and are of length at most d. We show that the problem of deciding whether an instance of d-SRTI admits a stable matching is NP-complete even if d=3. We also consider the "most stable" version of this problem and prove a strong inapproximability bound for the d=3 case. However for d=2 we show that the latter problem can be solved in polynomial time.Comment: short version appeared at SAGT 201

An Algorithm for Strong Stability in the Student-Project Allocation Problem With Ties

We study a variant of the Student-Project Allocation problem with lecturer preferences over Students where ties are allowed in the preference lists of students and lecturers (spa-st). We investigate the concept of strong stability in this context. Informally, a matching is strongly stable if there is no student and lecturer l such that if they decide to form a private arrangement outside of the matching via one of l’s proposed projects, then neither party would be worse off and at least one of them would strictly improve. We describe the first polynomial-time algorithm to find a strongly stable matching or report that no such matching exists, given an instance of spa-st. Our algorithm runs in O(m2) time, where m is the total length of the students’ preference lists

Approximability results for stable marriage problems with ties

We consider instances of the classical stable marriage problem in which persons may include ties in their preference lists. We show that, in such a setting, strong lower bounds hold for the approximability of each of the problems of finding an egalitarian, minimum regret and sex-equal stable matching. We also consider stable marriage instances in which persons may express unacceptable partners in addition to ties. In this setting, we prove that there are constants delta, delta' such that each of the problems of approximating a maximum and minimum cardinality stable matching within factors of delta, delta' (respectively) is NP-hard, under strong restrictions. We also give an approximation algorithm for both problems that has a performance guarantee expressible in terms of the number of lists with ties. This significantly improves on the best-known previous performance guarantee, for the case that the ties are sparse. Our results have applications to large-scale centralized matching schemes

Super-stability in the Student-Project Allocation Problem with Ties

The Student-Project Allocation problem with lecturer preferences over Students ( Open image in new window ) involves assigning students to projects based on student preferences over projects, lecturer preferences over students, and the maximum number of students that each project and lecturer can accommodate. This classical model assumes that preference lists are strictly ordered. Here, we study a generalisation of Open image in new window where ties are allowed in the preference lists of students and lecturers, which we refer to as the Student-Project Allocation problem with lecturer preferences over Students with Ties ( Open image in new window ). We investigate stable matchings under the most robust definition of stability in this context, namely super-stability. We describe the first polynomial-time algorithm to find a super-stable matching or to report that no such matching exists, given an instance of Open image in new window . Our algorithm runs in O(L) time, where L is the total length of all the preference lists. Finally, we present results obtained from an empirical evaluation of the linear-time algorithm based on randomly-generated Open image in new window instances. Our main finding is that, whilst super-stable matchings can be elusive, the probability of such a matching existing is significantly higher if ties are restricted to the lecturers’ preference lists