This book series provides an up-to-date and reliable source of information from fundamental knowledge to emerging research areas related to theoretic computer science. The target audiences cover researchers, engineers, scientists, students and professionals. PrefaceMatching problems involving preferences occur in widespread applications such as the assignment of children to schools, school-leavers to universities, junior doctors to hospitals, students to campus housing, kidney transplant patients to donors and so on. The common thread is that agents have preferences over the possible outcomes and the task is to find a matching (i.e., an assignment of the participants to one another) that is in some sense optimal with respect to these preferences. These problems are growing in importance in an era in which more and more elements of society are embracing diverse forms of electronic communication, and individuals are increasingly used to making choices via the internet. The ease by which preference information can now be collected has contributed to the growing tendency for matching processes to be centralised. Due to the typical size of applications (for example, in China, over 10 million students apply for admission to higher education annually through a centralised process), trying to construct optimal allocations manually (given a suitable definition of "optimal") is simply not feasible.Thus algorithms are required to automate the process of constructing optimal matchings. Again, due to the size of typical applications, the efficiency of the algorithms is of paramount importance. The notion of optimality is also a key consideration: many matching processes are conducted by publicly-funded organisations, and there is an increasing tendency for the decisions reached by these organisations to be scrutinised both in the media and by individuals through Freedom of Information requests, for example. Thus the algorithms need to construct matchings that are not just provably optimal, but also are seen to be "fair" by the agents involved.This book focuses on algorithmic aspects of matching problems involving preferences -our aim is to describe efficient (polynomial-time viii Preface algorithms) that produce optimal matchings (under many different notions of optimality) or to highlight complexity results that imply the nonexistence of such algorithms. We also describe some of the many applications in which these algorithms are used. Our interest, therefore, is in the underlying computational matching problems that arise in matching markets. The importance of this research area was recognised by the award, in 2012, of the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel (commonly known as the Nobel Prize in Economic Sciences) to Alvin Roth and Lloyd Shapley, who are both leading figures in the research area.The archetypal matching problem involving preferences is the celebrated Stable Marriage problem, first introduced by David Gale and Lloyd Shapley in 1962 [235]. The main contribution of this pa...

The Stable Marriage Problem and its many variants have been widely studied in the literature [6, 22, 15], partly because of the inherent appeal of the problem, partly because of the elegance of the associated structures and algorithms, and partly because of important practical applications, such as the National Resident Matching Program [20] and similar large-scale matching schemes. Here, we present the first comprehensive study of variants of the problem in which the preference lists of the participants are not necessarily complete and not necessarily totally ordered. We show that, under surprisingly restrictive assumptions, a number of these variants are hard, and hard to approximate. The key observation is that, in contrast to the case where preference lists are complete or strictly ordered (or both), a given problem instance may admit stable matchings of different sizes. In this setting, examples of problems that are hard are: finding a stable matching of maximum or minimum size, determining whether a given pair is stable-even if the indifference takes the form of ties on one side only, the ties are at the tails of lists, there is at most one tie per list, and each tie is of length 2; and finding, or approximating, both an 'egalitarian' and a 'minimum regret' stable matching. However, we give a 2-approximation algorithm for the problems of finding a stable matching of maximum or minimum size. We also discuss the significant implications of our results for practical matching schemes.

The achromatic number ψ(G) of a graph G = (V, E) is the maximum k such that V has a partition V 1 , V 2 ,. .. , V k into independent sets, the union of no pair of which is independent. Here we show that ψ(G) can be viewed as the maximum over all minimal elements of a partial order defined on the set of all colourings of G. We introduce a natural refinement of this partial order, giving rise to a new parameter, which we call the b-chromatic number, ϕ(G), of G. We prove that determining ϕ(G) is NP-hard for general graphs, but polynomial-time solvable for trees.

We study two generalised stable matching problems motivated by the current matching scheme used in the higher education sector in Hungary. The first problem is an extension of the College Admissions problem in which the colleges have lower quotas as well as the normal upper quotas. Here, we show that a stable matching may not exist and we prove that the problem of determining whether one does is NP-complete in general. The second problem is a different extension in which, as usual, individual colleges have upper quotas, but in addition, certain bounded subsets of colleges have common quotas smaller than the sum of their individual quotas. Again, we show that a stable matching may not exist and the related decision problem is NP-complete. On the other hand we prove that, when the bounded sets form a nested set system, a stable matching can be found by generalising, in non-trivial ways, both the applicant-oriented and college-oriented versions of the classical Gale-Shapley algorithm. Finally, we present an alternative view of this nested case using the concept of choice functions, and with the aid of a matroid model we establish some interesting structural results for this case.

We study the Student-Project Allocation problem (SPA), a generalisation of the classical Hospitals/Residents problem (HR). An instance of SPA involves a set of students, projects and lecturers. Each project is offered by a unique lecturer, and both projects and lecturers have capacity constraints. Students have preferences over projects, whilst lecturers have preferences over students. We present two optimal linear-time algorithms for allocating students to projects, subject to the preference and capacity constraints. In particular, each algorithm finds a stable matching of students to projects. Here, the concept of stability generalises the stability definition in the HR context. The stable matching produced by the first algorithm is simultaneously best-possible for all students, whilst the one produced by the second algorithm is simultaneously best-possible for all lecturers. We also prove some structural results concerning the set of stable matchings in a given instance of SPA. The SPA problem model that we consider is very general and has applications to a range of different contexts besides student-project allocation.