Maximum weight cycle packing in directed graphs, with application to kidney exchange programs

Centralized matching programs have been established in several countries to organize kidney exchanges between incompatible patient-donor pairs. At the heart of these programs are algorithms to solve kidney exchange problems, which can be modelled as cycle packing problems in a directed graph, involving cycles of length 2, 3, or even longer. Usually, the goal is to maximize the number of transplants, but sometimes the total benefit is maximized by considering the differences between suitable kidneys. These problems correspond to computing cycle packings of maximum size or maximum weight in directed graphs. Here we prove the APX-completeness of the problem of finding a maximum size exchange involving only 2-cycles and 3-cycles. We also present an approximation algorithm and an exact algorithm for the problem of finding a maximum weight exchange involving cycles of bounded length. The exact algorithm has been used to provide optimal solutions to real kidney exchange problems arising from the National Matching Scheme for Paired Donation run by NHS Blood and Transplant, and we describe practical experience based on this collaboration

The stable roommates problem with globally-ranked pairs

We introduce a restriction of the stable roommates problem in which roommate pairs are ranked globally. In contrast to the unrestricted problem, weakly stable matchings are guaranteed to exist, and additionally, they can be found in polynomial time. However, it is still the case that strongly stable matchings may not exist, and so we consider the complexity of finding weakly stable matchings with various desirable properties. In particular, we present a polynomial-time algorithm to find a rank-maximal (weakly stable) matching. This is the first generalization of an algorithm due to [Irving et al. 06] to a nonbipartite setting. Also, we describe several hardness results in an even more restricted setting for each of the problems of finding weakly stable matchings that are of maximum size, are egalitarian, have minimum regret, and admit the minimum number of weakly blocking pairs

Modeling Stable Matching Problems with Answer Set Programming

The Stable Marriage Problem (SMP) is a well-known matching problem first introduced and solved by Gale and Shapley (1962). Several variants and extensions to this problem have since been investigated to cover a wider set of applications. Each time a new variant is considered, however, a new algorithm needs to be developed and implemented. As an alternative, in this paper we propose an encoding of the SMP using Answer Set Programming (ASP). Our encoding can easily be extended and adapted to the needs of specific applications. As an illustration we show how stable matchings can be found when individuals may designate unacceptable partners and ties between preferences are allowed. Subsequently, we show how our ASP based encoding naturally allows us to select specific stable matchings which are optimal according to a given criterion. Each time, we can rely on generic and efficient off-the-shelf answer set solvers to find (optimal) stable matchings.Comment: 26 page

Size versus stability in the marriage problem

Given an instance I of the classical Stable Marriage problem with Incomplete preference lists (smi), a maximum cardinality matching can be larger than a stable matching. In many large-scale applications of smi, we seek to match as many agents as possible. This motivates the problem of finding a maximum cardinality matching in I that admits the smallest number of blocking pairs (so is “as stable as possible”). We show that this problem is NP-hard and not approximable within n1−ε, for any ε>0, unless P=NP, where n is the number of men in I. Further, even if all preference lists are of length at most 3, we show that the problem remains NP-hard and not approximable within δ, for some δ>1. By contrast, we give a polynomial-time algorithm for the case where the preference lists of one sex are of length at most 2. We also extend these results to the cases where (i) preference lists may include ties, and (ii) we seek to minimize the number of agents involved in a blocking pair

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

"Almost stable" matchings in the Roommates problem

An instance of the classical Stable Roommates problem (SR) need not admit a stable matching. This motivates the problem of finding a matching that is “as stable as possible”, i.e. admits the fewest number of blocking pairs. In this paper we prove that, given an SR instance with n agents, in which all preference lists are complete, the problem of finding a matching with the fewest number of blocking pairs is NP-hard and not approximable within n^{\frac{1}{2}-\varepsilon}, for any \varepsilon>0, unless P=NP. If the preference lists contain ties, we improve this result to n^{1-\varepsilon}. Also, we show that, given an integer K and an SR instance I in which all preference lists are complete, the problem of deciding whether I admits a matching with exactly K blocking pairs is NP-complete. By contrast, if K is constant, we give a polynomial-time algorithm that finds a matching with at most (or exactly) K blocking pairs, or reports that no such matching exists. Finally, we give upper and lower bounds for the minimum number of blocking pairs over all matchings in terms of some properties of a stable partition, given an SR instance I

Randomized Parameterized Algorithms for the Kidney Exchange Problem

In order to increase the potential kidney transplants between patients and their incompatible donors, kidney exchange programs have been created in many countries. In the programs, designing algorithms for the kidney exchange problem plays a critical role. The graph theory model of the kidney exchange problem is to find a maximum weight packing of vertex-disjoint cycles and chains for a given weighted digraph. In general, the length of cycles is not more than a given constant L (typically 2 L 5), and the objective function corresponds to maximizing the number of possible kidney transplants. In this paper, we study the parameterized complexity and randomized algorithms for the kidney exchange problem without chains from theory. We construct two different parameterized models of the kidney exchange problem for two cases L = 3 and L 3, and propose two randomized parameterized algorithms based on the random partitioning technique and the randomized algebraic technique, respectively