Solving Unweighted and Weighted Bipartite Matching Problems in Theory and Practice
Solving Unweighted and Weighted Bipartite Matching Problems in Theory and Practice

Author: J. Robert Kennedy

Publisher:

Published: 1995

Total Pages: 222

ISBN-13:

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Abstract: "The push-relabel method has been shown to be efficient for solving maximum flow and minimum cost flow problems in practice, and periodic global updates of dual variables have played an important role in the best implementations. Nevertheless, global updates had not been known to yield any theoretical improvement in running time. In this work, we study techniques for implementing push-relabel algorithms to solve bipartite matching and assignment problems. We show that global updates yield a theoretical improvement in the bipartite matching and assignment contexts, and we develop a suite of efficient cost-scaling push-relabel implementations to solve assignment problems. For bipartite matching, we show that a push-relabel algorithm using global updates runs in [formula] time (matching the best bound known) and performs worse by a factor of [square root of n] without the updates. We present a similar result for the assignment problem, for which an algorithm that assumes integer costs in the range [-C ..., C] runs in time O([square root of nm] log(nC)) (matching the best cost-scaling bound known). We develop cost-scaling push-relabel implementations that take advantage of the assignment problem's special structure, and compare our codes against the best codes from the literature. The results show that the push-relabel method is very promising for practical use."

Online Matching and Ad Allocation
Online Matching and Ad Allocation

Author: Aranyak Mehta

Publisher:

Published: 2013-10-01

Total Pages: 120

ISBN-13: 9781601987181

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Matching is a classic problem with a rich history and a significant impact on both the theory of algorithms and in practice. Recently, there has been a surge of interest in the online version of matching and its generalizations. This is due to the important new application domain of Internet advertising. The theory of online matching and allocation has played a critical role in designing algorithms for ad allocation. Online Matching and Ad Allocation surveys the key problems, models, and algorithms from online matchings, as well as their implication in the practice of ad allocation. It provides a classification of the problems in this area, an introduction into the techniques used, a glimpse into the practical impact, and ponders some of the open questions that will be of interest in the future. Matching continues to find core applications in diverse domains, and the advent of massive online and streaming data emphasizes the future applicability of the algorithms and techniques surveyed here. Online Matching and Ad Allocation is an ideal primer for anyone interested in matching, and particularly in the online version of the problem, in bipartite graphs.

Iterative Methods in Combinatorial Optimization
Iterative Methods in Combinatorial Optimization

Author: Lap Chi Lau

Publisher: Cambridge University Press

Published: 2011-04-18

Total Pages: 255

ISBN-13: 1139499394

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With the advent of approximation algorithms for NP-hard combinatorial optimization problems, several techniques from exact optimization such as the primal-dual method have proven their staying power and versatility. This book describes a simple and powerful method that is iterative in essence and similarly useful in a variety of settings for exact and approximate optimization. The authors highlight the commonality and uses of this method to prove a variety of classical polyhedral results on matchings, trees, matroids and flows. The presentation style is elementary enough to be accessible to anyone with exposure to basic linear algebra and graph theory, making the book suitable for introductory courses in combinatorial optimization at the upper undergraduate and beginning graduate levels. Discussions of advanced applications illustrate their potential for future application in research in approximation algorithms.

CATBox
CATBox

Author: Winfried Hochstättler

Publisher: Springer Science & Business Media

Published: 2010-03-16

Total Pages: 190

ISBN-13: 3642038220

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Graph algorithms are easy to visualize and indeed there already exists a variety of packages to animate the dynamics when solving problems from graph theory. Still it can be difficult to understand the ideas behind the algorithm from the dynamic display alone. CATBox consists of a software system for animating graph algorithms and a course book which we developed simultaneously. The software system presents both the algorithm and the graph and puts the user always in control of the actual code that is executed. In the course book, intended for readers at advanced undergraduate or graduate level, computer exercises and examples replace the usual static pictures of algorithm dynamics. For this volume we have chosen solely algorithms for classical problems from combinatorial optimization, such as minimum spanning trees, shortest paths, maximum flows, minimum cost flows, weighted and unweighted matchings both for bipartite and non-bipartite graphs. Find more information at http://schliep.org/CATBox/.

Research Trends in Combinatorial Optimization
Research Trends in Combinatorial Optimization

Author: William J. Cook

Publisher: Springer Science & Business Media

Published: 2008-11-07

Total Pages: 565

ISBN-13: 3540767967

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The editors and authors dedicate this book to Bernhard Korte on the occasion of his seventieth birthday. We, the editors, are happy about the overwhelming feedback to our initiative to honor him with this book and with a workshop in Bonn on November 3–7,2008.Althoughthiswouldbeareasontolookback,wewouldratherliketolook forward and see what are the interesting research directions today. This book is written by leading experts in combinatorial optimization. All - pers were carefully reviewed, and eventually twenty-three of the invited papers were accepted for this book. The breadth of topics is typical for the eld: combinatorial optimization builds bridges between areas like combinatorics and graph theory, submodular functions and matroids, network ows and connectivity, approximation algorithms and mat- matical programming, computational geometry and polyhedral combinatorics. All these topics are related, and they are all addressed in this book. Combi- torial optimization is also known for its numerous applications. To limit the scope, however, this book is not primarily about applications, although some are mentioned at various places. Most papers in this volume are surveys that provide an excellent overview of an activeresearcharea,butthisbookalsocontainsmanynewresults.Highlightingmany of the currently most interesting research directions in combinatorial optimization, we hope that this book constitutes a good basis for future research in these areas.