Paired and altruistic kidney donation in the UK: Algorithms and experimentation

We study the computational problem of identifying optimal sets of kidney exchanges in the UK. We show how to expand an integer programming-based formulation due to Roth et al. [2007] in order to model the criteria that constitute the UK definition of optimality. The software arising from this work has been used by the National Health Service Blood and Transplant to find optimal sets of kidney exchanges for their National Living Donor Kidney Sharing Schemes since July 2008. We report on the characteristics of the solutions that have been obtained in matching runs of the scheme since this time. We then present empirical results arising from experiments on the real datasets that stem from these matching runs, with the aim of establishing the extent to which the particular optimality criteria that are present in the UK influence the structure of the solutions that are ultimately computed. A key observation is that allowing four-way exchanges would be likely to lead to a moderate number of additional transplants

Modelling and Solving the Stable Marriage Problem Using Constraint Programming

We study the Stable Marriage problem (SM), which is a combinatorial problem that arises in many practical applications. We present two new models of an instance I of SM with n men and n women as an instance J of a Constraint Satisfaction Problem. We prove that establishing arc consistency in J yields the same structure as given by the established Extended Gale/Shapley algorithm for SM as applied to I. Consequently, a solution (stable matching) of I can be derived without search. Furthermore we show that, in both encodings, all stable matchings in I may be enumerated in a failure-free manner. Our first encoding is of O(n^3) complexity and is very natural, whilst our second model, of O(n^2) complexity (which is optimal), is a development of the Boolean encoding in [6], establishing a greater level of structure

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

Algorithmic aspects of stable matching problems

The Stable Marriage problem (SM), the Hospitals/Residents problem (HR) and the Stable Roommates problem (SR) are three classical stable matching problems that were first studied by Gale and Shapley in 1962. These problems have widespread practical application in centralised automated matching schemes, which assign applicants to posts based on preference lists and capacity constraints in both the UK and internationally. Within such schemes it is often the case that an agent's preference list may be incomplete, and agents may also be allowed to express indifference in the form of ties. In the presence of ties, three stability criteria can be defined, namely weak stability, strong stability and super-stability. In this thesis we consider stable matching problems from an algorithmic point of view. Some of the problems that we consider are derived from new stable matching models, whilst others are obtained from existing stable matching models involving ties and incomplete lists, with additional natural restrictions on the problem instance. Furthermore, we also explore the use of constraint programming with both SM and HR. We first study a new variant of the Student-Project Allocation problem in which each student ranks a set of acceptable projects in preference order and similarly each lecturer ranks his available projects in preference order. In this context, two stability definitions can be identified, namely weak stability and strong stability. We show that the problem of finding a maximum weakly stable matching is NP-hard. However, we describe two 2-approximation algorithms for this problem. Regarding strong stability, we describe a polynomial-time algorithm for finding such a matching or reporting that none exists. Next we investigate SM with ties and incomplete lists (SMTI), and HR with ties (HRT), where the length of each agent's list is subject to an upper bound. We present both polynomial-time algorithms and NP-hardness results for a range of problems that are derived from imposing upper bounds on the length of the lists on one or both sides. We also consider HRT, and SR with ties and incomplete lists (SRTI), where the preference lists of one or both sets of agents (as applicable) are derived from one or two master lists in which agents are ranked. For super-stability, in the case of each of HRT and SRTI with a master list, we describe a linear-time algorithm that simplifies the algorithm used in the general case. In the case of strong stability, for each of HRT and SRTI with a master list, we describe an algorithm that is faster than that for the general case. We also show that, given an instance I of SRTI with a master list, the problem of finding a weakly stable matching is polynomial-time solvable. However, we show that given such an I, the problem of finding a maximum weakly stable matching is NP-hard. Other new stable matching models that we study are the variants of SMTI and SRTI with symmetric preferences. In this context we consider two models that are derived from alternative ways of interpreting the rank of an agent in the presence of ties. For both models we show that deciding if a complete weakly stable matching exists is NP-complete. Then for one of the models we show that each of the problem of finding a minimum regret and an egalitarian weakly stable matching is NP-hard and that the problem of determining if a (man,woman) pair belongs to a weakly stable matching is NP-complete. We then describe algorithms for each of the problems of finding a super-stable matching and a strongly stable matching, or reporting that none exists, given instances of SRTI and HRT with symmetric preferences (regardless of how the ranks are interpreted). Finally, we use constraint programming techniques to model instances of SM and HR. We describe two encodings of SM in terms of a constraint satisfaction problem. The first model for SM is then extended to the case of HR. This encoding for HR is then extended to create a model for HRT under weak stability. Using this encoding we can obtain, with the aid of search, all the weakly stable matchings, given an instance of HRT

Stable Marriage with Ties and Bounded Length Preference Lists

We consider variants of the classical stable marriage problem in which preference lists may contain ties, and may be of bounded length. Such restrictions arise naturally in practical applications, such as centralised matching schemes that assign graduating medical students to their first hospital posts. In such a setting, weak stability is the most common solution concept, and it is known that weakly stable matchings can have different sizes. This motivates the problem of finding a maximum cardinality weakly stable matching, which is known to be NP-hard in general. We show that this problem is solvable in polynomial time if each man's list is of length at most 2 (even for women's lists that are of unbounded length). However if each man's list is of length at most 3, we show that the problem becomes NP-hard and not approximable within some d > 1, even if each woman's list is of length at most 4

A Constraint Programming Approach to the Hospitals / Residents Problem

An instance I of the Hospitals / Residents problem (HR) involves a set of residents (graduating medical students) and a set of hospitals, where each hospital has a given capacity. The residents have preferences for the hospitals, as do hospitals for residents. A solution of I is a stable matching, which is an assignment of residents to hospitals that respects the capacity conditions and preference lists in a precise way. In this paper we present constraint encodings for HR that give rise to important structural properties. We also present a computational study using both randomly-generated and real-world instances. Our study suggests that Constraint Programming is indeed an applicable technology for solving this problem, in terms of both theory and practice

Modelling and Solving the Stable Marriage Problem Using Constraint Programming

We study the Stable Marriage problem (SM), which is a combinatorial problem that arises in many practical applications. We present two new models of an instance I of SM with n men and n women as an instance J of a Constraint Satisfaction Problem. We prove that establishing arc consistency in J yields the same structure as given by the established Extended Gale/Shapley algorithm for SM as applied to I. Consequently, a solution (stable matching) of I can be derived without search. Furthermore we show that, in both encodings, all stable matchings in I may be enumerated in a failure-free manner. Our first encoding is of O(n^3) complexity and is very natural, whilst our second model, of O(n^2) complexity (which is optimal), is a development of the Boolean encoding in [6], establishing a greater level of structure

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, which motivates max-spa-p, the problem of finding maximum cardinality stable matching. We prove that max-spa-p is NP-hard and not approximable within δ, for some δ>1, unless P=NP. On the other hand, we give an approximation algorithm with a performance guarantee of 2 for max-spa-p

Stable marriage with ties and bounded length preference lists

We consider variants of the classical stable marriage problem in which preference lists may contain ties, and may be of bounded length. Such restrictions arise naturally in practical applications, such as centralised matching schemes that assign graduating medical students to their first hospital posts. In such a setting, weak stability is the most common solution concept, and it is known that weakly stable matchings can have different sizes. This motivates the problem of finding a maximum cardinality weakly stable matching, which is known to be NP-hard in general. We show that this problem is solvable in polynomial time if each man's list is of length at most 2 (even for women's lists that are of unbounded length). However if each man's list is of length at most 3, we show that the problem becomes NP-hard (even if each women's list is of length at most 3) and not approximable within some δ>1 (even if each woman's list is of length at most 4)

Hypoalbuminaemia is associated with mortality in patients undergoing cytoreductive nephrectomy

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/113110/1/bju12897.pd