The Strongly Stable Roommates Problem

An instance of the strongly stable roommates problem with incomplete lists and ties (SRTI) is an undirected non-bipartite graph G = (V,E), with an adjacency list being a linearly ordered list of ties, which are vertices equally good for a given vertex. Ties are disjoint and may contain one vertex. A matching M is a set of vertex-disjoint edges. An edge {x, y} in EM is a blocking edge for M if x is either unmatched or strictly prefers y to its current partner in M, and y is either unmatched or strictly prefers x to its current partner in M or is indifferent between them. A matching is strongly stable if there is no blocking edge with respect to it. We present an O(nm) time algorithm for computing a strongly stable matching, where we denote n = |V| and m = |E|. The best previously known solution had running time O(m^2) [Scott, 2005]. We also give a characterisation of the set of all strongly stable matchings. We show that there exists a partial order with O(m) elements representing the set of all strongly stable matchings, and we give an O(nm) algorithm for constructing such a representation. Our algorithms are based on a simple reduction to the bipartite version of the problem

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

Efficient algorithms for generalized Stable Marriage and Roommates problems

We consider a generalization of the Stable Roommates problem (SR), in which preference lists may be partially ordered and forbidden pairs may be present, denoted by SRPF. This includes, as a special case, a corresponding generalization of the classical Stable Marriage problem (SM), denoted by SMPF. By extending previous work of Feder, we give a two-step reduction from SRPF to 2-SAT. This has many consequences, including fast algorithms for a range of problems associated with finding "optimal" stable matchings and listing all solutions, given variants of SR and SM. For example, given an SMPF instance I, we show that there exists an O(m) "succinct" certificate for the unsolvability of I, an O(m) algorithm for finding all the super-stable pairs in I, an O(m+kn) algorithm for listing all the super-stable matchings in I, an O(m<sup>1.5</sup>) algorithm for finding an egalitarian super-stable matching in I, and an O(m) algorithm for finding a minimum regret super-stable matching in I, where n is the number of men, m is the total length of the preference lists, and k is the number of super-stable matchings in I. Analogous results apply in the case of SRPF

An Algorithm for the Maximum Weight Strongly Stable Matching Problem

An instance of the maximum weight strongly stable matching problem with incomplete lists and ties is an undirected bipartite graph G = (A cup B, E), with an adjacency list being a linearly ordered list of ties, which are vertices equally good for a given vertex. We are also given a weight function w on the set E. An edge (x, y) in E setminus M is a blocking edge for M if by getting matched to each other neither of the vertices x and y would become worse off and at least one of them would become better off. A matching is strongly stable if there is no blocking edge with respect to it. The goal is to compute a strongly stable matching of maximum weight with respect to w. We give a polyhedral characterisation of the problem and prove that the strongly stable matching polytope is integral. This result implies that the maximum weight strongly stable matching problem can be solved in polynomial time. Thereby answering an open question by Gusfield and Irving [Dan Gusfield and Robert W. Irving, 1989]. The main result of this paper is an efficient O(nm log{(Wn)}) time algorithm for computing a maximum weight strongly stable matching, where we denote n = |V|, m = |E| and W is a maximum weight of an edge in G. For small edge weights we show that the problem can be solved in O(nm) time. Note that the fastest known algorithm for the unweighted version of the problem has O(nm) runtime [Telikepalli Kavitha et al., 2007]. Our algorithm is based on the rotation structure which was constructed for strongly stable matchings in [Adam Kunysz et al., 2016]

Size versus truthfulness in the House Allocation problem

We study the House Allocation problem (also known as the Assignment problem), i.e., the problem of allocating a set of objects among a set of agents, where each agent has ordinal preferences (possibly involving ties) over a subset of the objects. We focus on truthful mechanisms without monetary transfers for finding large Pareto optimal matchings. It is straightforward to show that no deterministic truthful mechanism can approximate a maximum cardinality Pareto optimal matching with ratio better than 2. We thus consider randomised mechanisms. We give a natural and explicit extension of the classical Random Serial Dictatorship Mechanism (RSDM) specifically for the House Allocation problem where preference lists can include ties. We thus obtain a universally truthful randomised mechanism for finding a Pareto optimal matching and show that it achieves an approximation ratio of $\frac{e}{e-1}$. The same bound holds even when agents have priorities (weights) and our goal is to find a maximum weight (as opposed to maximum cardinality) Pareto optimal matching. On the other hand we give a lower bound of $\frac{18}{13}$ on the approximation ratio of any universally truthful Pareto optimal mechanism in settings with strict preferences. In the case that the mechanism must additionally be non-bossy with an additional technical assumption, we show by utilising a result of Bade that an improved lower bound of $\frac{e}{e-1}$ holds. This lower bound is tight since RSDM for strict preference lists is non-bossy. We moreover interpret our problem in terms of the classical secretary problem and prove that our mechanism provides the best randomised strategy of the administrator who interviews the applicants.Comment: To appear in Algorithmica (preliminary version appeared in the Proceedings of EC 2014

Stable marriage and roommates problems with restricted edges: complexity and approximability

In the Stable Marriage and Roommates problems, a set of agents is given, each of them having a strictly ordered preference list over some or all of the other agents. A matching is a set of disjoint pairs of mutually acceptable agents. If any two agents mutually prefer each other to their partner, then they block the matching, otherwise, the matching is said to be stable. We investigate the complexity of finding a solution satisfying additional constraints on restricted pairs of agents. Restricted pairs can be either forced or forbidden. A stable solution must contain all of the forced pairs, while it must contain none of the forbidden pairs. Dias et al. (2003) gave a polynomial-time algorithm to decide whether such a solution exists in the presence of restricted edges. If the answer is no, one might look for a solution close to optimal. Since optimality in this context means that the matching is stable and satisfies all constraints on restricted pairs, there are two ways of relaxing the constraints by permitting a solution to: (1) be blocked by as few as possible pairs, or (2) violate as few as possible constraints n restricted pairs. Our main theorems prove that for the (bipartite) Stable Marriage problem, case (1) leads to View the MathML source-hardness and inapproximability results, whilst case (2) can be solved in polynomial time. For non-bipartite Stable Roommates instances, case (2) yields an View the MathML source-hard but (under some cardinality assumptions) 2-approximable problem. In the case of View the MathML source-hard problems, we also discuss polynomially solvable special cases, arising from restrictions on the lengths of the preference lists, or upper bounds on the numbers of restricted pairs

Efficient algorithms for bipartite matching problems with preferences

Matching problems involve a set of participants, where each participant has a capacity and a subset of the participants rank a subset of the others in order of preference (strictly or with ties). Matching problems are motivated in practice by large-scale applications, such as automated matching schemes, which assign participants together based on their preferences over one another. This thesis focuses on bipartite matching problems in which there are two disjoint sets of participants (such as medical students and hospitals). We present a range of efficient algorithms for finding various types of optimal matchings in the context of these problems. Our optimality criteria involve a diverse range of concepts that are alternatives to classical stability. Examples include so-called popular and Pareto optimal matchings, and also matchings that are optimal with respect to their profile (the number of participants obtaining their first choice, second choice and so on). The first optimality criterion that we study is the notion of a Pareto optimal matching, a criterion that economists regard as a fundamental property to be satisfied by an optimal matching. We present the first algorithmic results on Pareto optimality for the Capacitated House Allocation problem (CHA), which is a many-to-one variant of the classical House Allocation problem, as well as for the Hospitals-Residents problem (HR), a generalisation of the classical Stable Marriage problem. For each of these problems, we obtain a characterisation of Pareto optimal matchings, and then use this to obtain a polynomial-time algorithm for finding a maximum Pareto optimal matching. The next optimality criterion that we study is the notion of a popular matching. We study popular matchings in CHA and present a polynomial-time algorithm for finding a maximum popular matching or reporting that none exists, given any instance of CHA. We extend our findings to the case in CHA where preferences may contain ties (CHAT) by proving the extension of a well-known result in matching theory to the capacitated bipartite graph case, and using this to obtain a polynomial-time algorithm for finding a maximum popular matching, or reporting that none exists. We next study popular matchings in the Weighted Capacitated House Allocation problem (WCHA), which is a variant of CHA where the agents have weights assigned to them. We identify a structure in the underlying graph of the problem that singles out those edges that cannot belong to a popular matching. We then use this to construct a polynomial-time algorithm for finding a maximum popular matching or reporting that none exists, for the case where preferences are strict. We then study popular matchings in a variant of the classical Stable Marriage problem with Ties and Incomplete preference lists (SMTI), where preference lists are symmetric. Here, we provide the first characterisation results on popular matchings in the bipartite setting where preferences are two-sided, which can either lead to a polynomial-time algorithm for solving the problem or help establish that it is NP-complete. We also provide the first algorithm for testing if a matching is popular in such a setting. The remaining optimality criteria that we study involve profile-based optimal matchings. We define three versions of what it means for a matching to be optimal based on its profile, namely so-called greedy maximum, rank-maximal and generous maximum matchings. We study each of these in the context of CHAT and the Hospitals-Residents problem with Ties (HRT). For each problem model, we give polynomial-time algorithms for finding a greedy maximum, a rank-maximal and a generous maximum matching

Counting Perfect Matchings and the Switch Chain

We examine the problem of exactly or approximately counting all perfect matchings in hereditary classes of nonbipartite graphs. In particular, we consider the switch Markov chain of Diaconis, Graham, and Holmes. We determine the largest hereditary class for which the chain is ergodic, and define a large new hereditary class of graphs for which it is rapidly mixing. We go on to show that the chain has exponential mixing time for a slightly larger class. We also examine the question of ergodicity of the switch chain in an arbitrary graph. Finally, we give exact counting algorithms for three classes