The Stable Marriage Problem

This book probes the stable marriage problem and its variants as a rich source of problems and ideas that illustrate both the design and analysis of efficient algorithms. It covers the most recent structural and algorithmic work on stable matching problems, simplifies and unifies many earlier proofs, strengthens several earlier results, and presents new results and more efficient algorithms. The authors develop the structure of the set of stable matchings in the stable marriage problem in a more general and algebraic context than has been done previously; they discuss the problem's structure in terms of rings of sets, which allows many of the most useful features to be seen as features of a more general set of problems. The relationship between the structure of the stable marriage problem and the more general stable roommates problem is demonstrated, revealing many commonalities. The results the authors obtain provide an algorithmic response to the practical, and political, problems created by the asymmetry inherent in the Gale Shapley solutions, leading to alternative methods and better compromises than are provided by the Gale Shapley method. And, in contrast to Donald Knuth's earlier work which primarily focused on the application of mathematics to the analysis of algorithms, this book illustrates the productive and almost inseparable relationship between mathematical insight and the design of efficient algorithms. Dan Gusfield is Associate Professor of Computer Science at the University of California, Davis. Robert W. Irving is Senior Lecturer in Computing Science at the University of Glasgow. The Stable Marriage Problem is included in the Foundations of Computing Series, edited by Michael Garey and Albert Meyer.

'This is a very stimulating book!' - N. G. de Bruijn. 'This short book will provide extremely enjoyable reading to anyone with an interest in discrete mathematics and algorithm design' - ""Mathematical Reviews"". 'This book is an excellent (and enjoyable) means of sketching a large area of computer science for specialists in other fields: It requires little previous knowledge, but expects of the reader a degree of mathematical facility and a willingness to participate. It is really neither a survey nor an introduction; rather, it is a paradigm, a fairly complete treatment of a single example used as a synopsis of a larger subject' - ""SIGACT News"". 'Anyone would enjoy reading this book. If one had to learn French first, it would be worth the effort!' - ""Computing Reviews"". The above citations are taken from reviews of the initial French version of this text - a series of seven expository lectures that were given at the University of Montreal in November of 1975.The book uses the appealing theory of stable marriage to introduce and illustrate a variety of important concepts and techniques of computer science and mathematics: data structures, control structures, combinatorics, probability, analysis, algebra, and especially the analysis of algorithms. The presentation is elementary, and the topics are interesting to nonspecialists. The theory is quite beautiful and developing rapidly. Exercises with answers, an annotated bibliography, and research problems are included.The text would be appropriate as supplementary reading for undergraduate research seminars or courses in algorithmic analysis and for graduate courses in combinatorial algorithms, operations research, economics, or analysis of algorithms. Donald E. Knuth is one of the most prominent figures of modern computer science. His works in ""The Art of Computer Programming"" are classic. He is also renowned for his development of TeX and METAFONT. In 1996, Knuth won the prestigious Kyoto Prize, considered to be the nearest equivalent to a Nobel Prize in computer science.

Matching problems with preferences are all around us OCo they arise when agents seek to be allocated to one another on the basis of ranked preferences over potential outcomes. Efficient algorithms are needed for producing matchings that optimise the satisfaction of the agents according to their preference lists.In recent years there has been a sharp increase in the study of algorithmic aspects of matching problems with preferences, partly reflecting the growing number of applications of these problems worldwide. This book describes the most important results in this area, providing a timely update to The Stable Marriage Problem: Structure and Algorithms (D Gusfield and R W Irving, MIT Press, 1989) in connection with stable matching problems, whilst also broadening the scope to include matching problems with preferences under a range of alternative optimality criteria."

The papers contained in this volume were presented at the fourth edition of the IFIP International Conference on Theoretical Computer Science (IFIP TCS), held August 23-24, 2006 in Santiago, Chile. They were selected from 44 pa pers submitted from 17 countries in response to the call for papers. A total of 16 submissions were accepted as full papers, yielding an acceptance rate of about 36%. Papers sohcited for IFIP TCS 2006 were meant to constitute orig inal contributions in two general areas: Algorithms, Complexity and Models of Computation; and Logic, Semantics, Specification and Verification. The conference also included six invited presentations: Marcelo Arenas (P- tificia Universidad Catolica de Chile, Chile), Jozef Gruska (Masaryk University, Czech Republic), Claudio Gutierrez (Universidad de Chile, Chile), Marcos Kiwi (Universidad de Chile, Chile), Nicola Santoro (Carleton University, Canada), and Mihalis Yannakakis (Columbia University, USA). The abstracts of those presentations are included in this volume. In addition, Jozef Gruska and Nicola Santoro accepted our invitation to write full papers related to their talks. Those two surveys are included in the present volume as well. TCS is a biannual conference. The first edition was held in Sendai (Japan, 2000), followed by Montreal (Canada, 2002) and Toulouse (France, 2004).

In 1962, David Gale and Lloyd Shapley proved that, for any equal number of men and women, and each man and each woman give a strict order of preference to the opposite sex, they can always solve for at least one stable marriage solution. What they didn't tell us is how to find all stable solutions. In this paper, I will demonstrate how to find every stable solution to a stable marriage problem. And I will discover the expected number of the stable matching for the stable marriage problem. Moreover, I will use the scoring system to quantify how much a man or a woman prefer the opposite sex, denoted as h̀̀appiness score''. Then we can use the scoring system to present every matching on the $xy$-plane and look at the distribution of matchings.

One of Springer’s renowned Major Reference Works, this awesome achievement provides a comprehensive set of solutions to important algorithmic problems for students and researchers interested in quickly locating useful information. This first edition of the reference focuses on high-impact solutions from the most recent decade, while later editions will widen the scope of the work. All entries have been written by experts, while links to Internet sites that outline their research work are provided. The entries have all been peer-reviewed. This defining reference is published both in print and on line.

This book includes a set of rigorously reviewed world-class manuscripts addressing and detailing state-of-the-art research projects in the areas of Computing Sciences, Software Engineering and Systems. The book presents selected papers from the conference proceedings of the International Conference on Systems, Computing Sciences and Software Engineering (SCSS 2006). All aspects of the conference were managed on-line.