An Integer Programming Approach to the Student-Project Allocation Problem with Preferences over Projects

The Student-Project Allocation problem with preferences over Projects (SPA-P) involves sets of students, projects and lecturers, where the students and lecturers each have preferences over the projects. In this context, we typically seek a stable matching of students to projects (and lecturers). However, these stable matchings can have different sizes, and the problem of finding a maximum stable matching (MAX-SPA-P) is NP-hard. There are two known approximation algorithms for MAX-SPA-P, with performance guarantees of 2 and 32 . In this paper, we describe an Integer Programming (IP) model to enable MAX-SPA-P to be solved optimally. Following this, we present results arising from an empirical analysis that investigates how the solution produced by the approximation algorithms compares to the optimal solution obtained from the IP model, with respect to the size of the stable matchings constructed, on instances that are both randomly-generated and derived from real datasets. Our main finding is that the 32 -approximation algorithm finds stable matchings that are very close to having maximum cardinality

Solving stable matching problems using answer set programming

Since the introduction of the stable marriage problem (SMP) by Gale and Shapley (1962), several variants and extensions have been investigated. While this variety is useful to widen the application potential, each variant requires a new algorithm for finding the stable matchings. To address this issue, we propose an encoding of the SMP using answer set programming (ASP), which can straightforwardly be adapted and extended to suit the needs of specific applications. The use of ASP also means that we can take advantage of highly efficient off-the-shelf solvers. To illustrate the flexibility of our approach, we show how our ASP encoding naturally allows us to select optimal stable matchings, i.e. matchings that are optimal according to some user-specified criterion. To the best of our knowledge, our encoding offers the first exact implementation to find sex-equal, minimum regret, egalitarian or maximum cardinality stable matchings for SMP instances in which individuals may designate unacceptable partners and ties between preferences are allowed. This paper is under consideration in Theory and Practice of Logic Programming (TPLP).Comment: Under consideration in Theory and Practice of Logic Programming (TPLP). arXiv admin note: substantial text overlap with arXiv:1302.725

Student-Project Allocation with Preferences over Projects

We study the problem of allocating students to projects, where both students and lecturers have preferences over projects, and both projects and lecturers have capacities. In this context we seek a stable matching of students to projects, which respects these preference and capacity constraints. Here, the stability definition generalises the corresponding notion in the context of the classical Hospitals / Residents problem. We show that stable matchings can have different sizes, and the problem of finding a maximum cardinality stable matching is NP-hard, though approximable within a factor of 2

An 8/5 approximation algorithm for a hard variant of stable marriage

When ties and incomplete preference lists are permitted in the Stable Marriage problem, stable matchings can have different sizes. The problem of finding a maximum cardinality stable matching in this context is NP-hard, even under very severe restrictions on the number, size and position of ties. In this paper, we describe a polynomial-time 8/5-approximation algorithm for a variant in which ties are on one side only and at the end of the preference lists. This variant is motivated by important applications in large scale centralized matching schemes

Fixed cardinality stable sets

Given an undirected graph G=(V,E) and a positive integer k in {1, ..., |V|}, we initiate the combinatorial study of stable sets of cardinality exactly k in G. Our aim is to instigate the polyhedral investigation of the convex hull of fixed cardinality stable sets, inspired by the rich theory on the classical structure of stable sets. We introduce a large class of valid inequalities to the natural integer programming formulation of the problem. We also present simple combinatorial relaxations based on computing maximum weighted matchings, which yield dual bounds towards finding minimum-weight fixed cardinality stable sets, and particular cases which are solvable in polynomial time.publishedVersio

Approximation algorithms for hard variants of the stable marriage and hospitals/residents problems

When ties and incomplete preference lists are permitted in the Stable Marriage and Hospitals/Residents problems, stable matchings can have different sizes. The problem of finding a maximum cardinality stable matching in this context is known to be NP-hard, even under very severe restrictions on the number, size and position of ties. In this paper, we describe polynomial-time 5/3-approximation algorithms for variants of these problems in which ties are on one side only and at the end of the preference lists. The particular variant is motivated by important applications in large scale centralised matching schemes

Finding large stable matchings

When ties and incomplete preference lists are permitted in the stable marriage and hospitals/residents problems, stable matchings can have different sizes. The problem of finding a maximum cardinality stable matching in this context is known to be NP-hard, even under very severe restrictions on the number, size, and position of ties. In this article, we present two new heuristics for finding large stable matchings in variants of these problems in which ties are on one side only. We describe an empirical study involving these heuristics and the best existing approximation algorithm for this problem. Our results indicate that all three of these algorithms perform significantly better than naive tie-breaking algorithms when applied to real-world and randomly-generated data sets and that one of the new heuristics fares slightly better than the other algorithms, in most cases. This study, and these particular problem variants, are motivated by important applications in large-scale centralized matching schemes