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

Fair and large stable matchings in the stable marriage and student-project allocation problems

In this thesis, we present new algorithmic and complexity results for specific matching problems involving preferences. In particular we study the Stable Marriage problem (SM) and the Student-Project Allocation problem (SPA) and their variants. A matching in these scenarios is an allocation of men to women (SM) or students to projects (SPA). Primarily we are interested in finding matchings that are stable. A stable matching is a matching that admits no blocking pair, which is a pair of agents (not already allocated together) who would rather deviate from the given matching and become assigned to each other. In addition to stability, other objectives may be applied. We focus on finding either fair or large stable matchings in SM and SPA. In the Stable Marriage problem with Incomplete lists (SMI), the rank of a matched man or woman, with respect to a matching, is the position of their assigned partner on their preference list. The degree of the set of men in a matching is the rank of a worst-off man, over all matched men. A similar definition exists for the set of women. The cost of the set of men in a matching is sum of ranks of all matched men. Again a similar definition exists for the set of women. We introduce the following degree-based definitions of fairness in SMI. A stable matching is regret-equal if it minimises the difference in degree between the set of men and the set of women, over all stable matchings. Additionally, a stable matching is min-regret sum if it minimises the sum of the degree of men and the degree of women, over all stable matchings. We present polynomial-time algorithms to find these types of fair stable matchings, given an instance of SMI, and perform experiments to both test the performance of two algorithms to find a regret-equal stable matching, and to compare properties of several other types of fair stable matchings over both degree- and cost-based measures. Also in SMI, we investigate fairness in the form of profile-based stable matchings. The profile of a matching is a vector of integers, where the ith vector element indicates the number of agents assigned to their ith-choice partner. A stable matching is rank-maximal if its profile is lexicographically maximum taken over all stable matchings. A stable matching is generous if its reverse profile is lexicographically minimum taken over all stable matchings. A polynomial-time algorithm exists to find a rank-maximal stable matching, using weights that are exponential in the number of men or women [32]. We adapt this algorithm to work with polynomially-bounded weight vectors, and show using randomly-generated instances that our approach is significantly less costly in terms of space. Further experiments are carried out to compare these profile-based optimal matchings over several cost- and profile-based fairness properties. We additionally show that in the Stable Roommates problem, each of the problems of finding a rank-maximal stable matching or a generous stable matching is NP-hard. In the Student-Project Allocation problem with lecturer preferences over Students including Ties (SPA-ST) we study the problem of finding large stable matchings. A 3/2-approximation algorithm exists to find a maximum-sized stable matching in HRT, and we extend this to the SPA-ST case, developing a linear-time 3/2-approximation algorithm for the problem of finding a maximum-sized stable matching in SPA-ST. We test the performance of our approximation algorithm using the implementation of a new Integer Programming (IP) model that finds a maximum-sized stable matching, and show that in practice, our approximation algorithm produces stable matchings of size far closer to optimal than the 3/2 bound. Additionally, we give an example to show that this bound is tight. Finally, we look at fairness in the context of the Student-Project Allocation problem with lecturer preferences over Students including Ties and Lecturer targets (SPA-STL), an extension to SPA-ST in which each lecturer has an associated target, indicating their preferred number of allocations (or the number of allocations preferred by the matching scheme administrator). We first investigate load balancing without the presence of stability. A load-max-balanced matching is a matching in which the maximum difference between a lecturer’s target and their number of allocations is minimised. A load-sum-balanced matching is a matching in which the sum of differences between all lecturers’ targets and their number of allocations is minimised. Finally, a load-balanced matching is a matching that is both load-max-balanced and load-sum-balanced. We provide new polynomial-time algorithms to find matchings of these types. Additionally we show that in the presence of stability, each of the problems of finding a stable matching that is load-max-balanced, load-sum-balanced or load-balanced is NP-hard. Finally, we present new IP models for finding such types of optimal stable matching

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

Matchings with lower quotas: Algorithms and complexity

We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph G=(A∪˙P,E)G=(A∪˙P,E) with weights on the edges in E, and with lower and upper quotas on the vertices in P. We seek a maximum weight many-to-one matching satisfying two sets of constraints: vertices in A are incident to at most one matching edge, while vertices in P are either unmatched or they are incident to a number of matching edges between their lower and upper quota. This problem, which we call maximum weight many-to-one matching with lower and upper quotas (WMLQ), has applications to the assignment of students to projects within university courses, where there are constraints on the minimum and maximum numbers of students that must be assigned to each project. In this paper, we provide a comprehensive analysis of the complexity of WMLQ from the viewpoints of classical polynomial time algorithms, fixed-parameter tractability, as well as approximability. We draw the line between NPNP-hard and polynomially tractable instances in terms of degree and quota constraints and provide efficient algorithms to solve the tractable ones. We further show that the problem can be solved in polynomial time for instances with bounded treewidth; however, the corresponding runtime is exponential in the treewidth with the maximum upper quota umaxumax as basis, and we prove that this dependence is necessary unless FPT=W[1]FPT=W[1]. The approximability of WMLQ is also discussed: we present an approximation algorithm for the general case with performance guarantee umax+1umax+1, which is asymptotically best possible unless P=NPP=NP. Finally, we elaborate on how most of our positive results carry over to matchings in arbitrary graphs with lower quotas

Matching under Preferences

Matching theory studies how agents and/or objects from different sets can be matched with each other while taking agents\u2019 preferences into account. The theory originated in 1962 with a celebrated paper by David Gale and Lloyd Shapley (1962), in which they proposed the Stable Marriage Algorithm as a solution to the problem of two-sided matching. Since then, this theory has been successfully applied to many real-world problems such as matching students to universities, doctors to hospitals, kidney transplant patients to donors, and tenants to houses. This chapter will focus on algorithmic as well as strategic issues of matching theory. Many large-scale centralized allocation processes can be modelled by matching problems where agents have preferences over one another. For example, in China, over 10 million students apply for admission to higher education annually through a centralized process. The inputs to the matching scheme include the students\u2019 preferences over universities, and vice versa, and the capacities of each university. The task is to construct a matching that is in some sense optimal with respect to these inputs. Economists have long understood the problems with decentralized matching markets, which can suffer from such undesirable properties as unravelling, congestion and exploding offers (see Roth and Xing, 1994, for details). For centralized markets, constructing allocations by hand for large problem instances is clearly infeasible. Thus centralized mechanisms are required for automating the allocation process. Given the large number of agents typically involved, the computational efficiency of a mechanism's underlying algorithm is of paramount importance. Thus we seek polynomial-time algorithms for the underlying matching problems. Equally important are considerations of strategy: an agent (or a coalition of agents) may manipulate their input to the matching scheme (e.g., by misrepresenting their true preferences or underreporting their capacity) in order to try to improve their outcome. A desirable property of a mechanism is strategyproofness, which ensures that it is in the best interests of an agent to behave truthfully

The Student-Project Allocation Problem: structure and algorithms

In this thesis we study the Student-Project Allocation problem (SPA), which is a matching problem based on the allocation of students to projects and lecturers. Students have preferences over projects, where each project is offered by one lecturer; whilst lecturers have preferences over students, or over the projects that they offer. We seek stable matchings of students to projects, which guarantee that no student and lecturer have an incentive to deviate from the matching by forming a private arrangement involving some project. We present new structural and algorithmic results for four problems related to SPA . We begin by characterising the stable matchings in an instance of the Student-Project Allocation problem with Lecturer preferences over Students (SPA-S) where the preferences are strictly ordered, in the special case that for each student in the instance, all of the projects in her preference list are offered by different lecturers. We achieve this characterisation by showing that, under this restriction, the set of stable matchings in an instance of SPA-S is a distributive lattice with respect to a natural dominance relation. Next, we study a variant of SPA - S where the preferences may involve ties — the Student- Project Allocation problem with Lecturer preferences over Students with Ties (SPA-ST). The presence of ties in the preference lists gives rise to three different concepts of stability, namely, weak stability, strong stability, and super-stability. We investigate stable matchings under the super-stability (respectively strong stability) concept. We present the first polynomial-time algorithm to find a super-stable (respectively strongly stable) matching or to report that no such matching exists, given an instance of SPA-ST . We also prove some structural results concerning the set of super-stable (respectively strongly stable) matchings in a given instance of SPA - ST . Further, we present results obtained from an empirical evaluation of our algorithms based on randomly-generated SPA-ST instances. Moving away from variants of SPA with lecturer preferences over students, we study the Student-Project Allocation problem with lecturer preferences over Projects (SPA-P). In this context it is known that stable matchings can have different sizes and the problem of finding a maximum size stable matching, denoted MAX-SPA-P , is NP-hard. There are two known approximation algorithms for MAX-SPA-P , with performance guarantees 2 and 3/2 . We show that MAX-SPA-P is polynomial-time solvable if there is only one lecturer involved, and NP-hard to approximate within some constant c > 1 if there are two lecturers involved. We also show that this problem remains NP-hard if each preference list is of length at most 3, with an arbitrary number of lecturers. We then describe an Integer Programming (IP) model to enable MAX-SPA-P to be solved optimally in the general case. Following this, we present results arising from an empirical evaluation that investigates how the solutions produced by the approximation algorithms compare to optimal solutions 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

Student-project allocation with preferences over projects: algorithmic and experimental results

We study the Student-Project Allocation problem with lecturer preferences over Projects (spa-p). In this context it is known that stable matchings can have different sizes and the problem of finding a maximum size stable matching is NP-hard. There are two known approximation algorithms for max-spa-p, with performance guarantees 2 and . We show that max-spa-p is polynomial-time solvable if there is only one lecturer involved, and NP-hard to approximate within some constant if there are two lecturers involved. We also show that this problem remains NP-hard if each preference list is of length at most 3, with an arbitrary number of lecturers. We then describe an Integer Programming (IP) model to enable max-spa-p to be solved optimally in the general case. Following this, we present results arising from an empirical evaluation that investigates how the solutions produced by the approximation algorithms compare to optimal solutions 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

Matchings with lower quotas : algorithms and complexity

We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph G=(A∪˙P,E) with weights on the edges in E, and with lower and upper quotas on the vertices in P. We seek a maximum weight many-to-one matching satisfying two sets of constraints: vertices in A are incident to at most one matching edge, while vertices in P are either unmatched or they are incident to a number of matching edges between their lower and upper quota. This problem, which we call maximum weight many-to-one matching with lower and upper quotas (WMLQ), has applications to the assignment of students to projects within university courses, where there are constraints on the minimum and maximum numbers of students that must be assigned to each project. In this paper, we provide a comprehensive analysis of the complexity of WMLQ from the viewpoints of classical polynomial time algorithms, fixed-parameter tractability, as well as approximability. We draw the line between NP-hard and polynomially tractable instances in terms of degree and quota constraints and provide efficient algorithms to solve the tractable ones. We further show that the problem can be solved in polynomial time for instances with bounded treewidth; however, the corresponding runtime is exponential in the treewidth with the maximum upper quota umax as basis, and we prove that this dependence is necessary unless FPT=W[1]. The approximability of WMLQ is also discussed: we present an approximation algorithm for the general case with performance guarantee umax+1, which is asymptotically best possible unless P=NP. Finally, we elaborate on how most of our positive results carry over to matchings in arbitrary graphs with lower quotas