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lau l iterative methods combinatorial optimization 2011 rep

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Textbook in PDF format As teachers and students of combinatorial optimization, we have often looked for material that illustrates the elegance of classical results on matchings, trees, matroids, and flows, but also highlights methods that have continued application. With the advent of approximation algorithms, some techniques from exact optimization such as the primal-dual method have indeed proven their staying power and versatility. In this book, we describe what we believe is a simple and powerful method that is iterative in essence and useful in a variety of settings. The core of the iterative methods we describe relies on a fundamental result in linear algebra that the row rank and column rank of a real matrix are equal. This seemingly elementary fact allows us via a counting argument to provide an alternate proof of the previously mentioned classical results; the method is constructive and the resulting algorithms are iterative with the correctness proven by induction. Furthermore, these methods generalize to accommodate a variety of additional constraints on these classical problems that render them NP-hard – a careful adaptation of the iterative method leads to very effective approximation algorithms for these cases. Our goal in this book has been to highlight the commonality and uses of this method and convince the readers of the generality and potential for future applications. We have used an elementary presentation style that should be accessible to anyone with introductory college mathematics exposure in linear algebra and basic graph theory. Whatever advanced material in these areas we require, we develop from scratch along the way. Some basic background on approximation algorithms such as is provided in the various books and surveys available on this subject will be useful in appreciating the power of the results we prove in this area. Other than the basic definition of an approximation algorithm and the understanding of polynomial-time complexity, no further technical background is required from this typically more advanced subject. An important secondary goal of the book is to provide a framework and material for introductory courses in combinatorial optimization at the upper-class undergraduate and beginning graduate levels. We hope the common approach across the chapters gives a comprehensive way to introduce these topics for the first time. The more advanced applications are useful illustrations for graduate students of their potential for future application in their research. Preliminaries Matching and vertex cover in bipartite graphs Spanning trees Matroids Arbor escence and rooted connectivity Submodular flows and applications Network matrices Matchings Network Constrained optimization problems Cut problems Iterative relaxation: Early and recent examples

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lau l iterative methods combinatorial optimization 2011 rep