ebook img

Advances in Combinatorial Optimization: Linear Programming Formulations of the Traveling Salesman and Other Hard Combinatorial Optimization Problems PDF

154 Pages·2016·7.61 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Advances in Combinatorial Optimization: Linear Programming Formulations of the Traveling Salesman and Other Hard Combinatorial Optimization Problems

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Diaby, Moustapha. Advances in combinatorial optimization : linear programming formulations of the traveling salesman and other hard combinatorial optimization problems / Moustapha Diaby (University of Connecticut, USA), Mark H. Karwan (University at Buffalo, The State University of New York, USA). pages cm Includes bibliographical references. ISBN 978-9814704878 (hardback : alk. paper) 1. Combinatorial optimization. 2. Mathematical optimization. I. Karwan, Mark H., 1951–II. Title. QA402.5.D524 2015 519.6'4—dc23 2015026035 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2016 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. In-house Editors: Amanda Yun/Dipasri Sardar Typeset by Stallion Press Email: [email protected] Printed in Singapore “Hâtez-vous lentement, et sans perdre courage, Vingt fois sur le métier remettez votre ouvrage, Polissez-le sans cesse, et le repolissez, Ajoutez quelquefois, et souvent effacez.” Nicolas Boileau-Despréaux Contents About the Authors Preface Acknowledgments Chapter 1. Introduction 1. Overview 2. Overview of Traditional Formulations of the TSP 3. Basic Notations, Definitions, and Assumptions for the Proposed Modeling Chapter 2. Basic IP Model Using the TSP 1. Introduction 2. The “Alternate TSP Polytope”: A Non-Exponential Abstraction of TSP Tours 3. “TSP Paths”: Path Representation of TSP Tours 4. Intuition of the LP Modeling of TSP Paths 5. Integer Programming (IP) Model of TSP Paths 6. Structure of the IP Polytope Chapter 3. Basic LP Model Using the TSP 1. Introduction 2. General Algebraic Characterizations of the LP Polytope 3. “Flow” Structure of the LP Polytope 4. Integrality of the LP Polytope 5. Linear Cost Function for the TSP Paths Chapter 4. Generic LP Modeling for COPs 1. Introduction 2. Unified Description and Classification of COPs 3. Generic Bipartite Network Flow-Based Model of SCCOP Solutions 4. Generic Flow Graphs (GFG) 5. Overall LP Models for SCCOPs Chapter 5. Non-Symmetry of the Basic (TSP) Model 1. Introduction 2. Non-Symmetry of the Basic Model 3. Non-Symmetry of “Complexes” of the Basic Model Chapter 6. Non-Applicability of Extended Formulations Theory 1. Introduction 2. Background Overview 3. Ill-Definition Condition for EFs 4. Redundancy Matters for Polytopes Stated in Independent Spaces Chapter 7. Illustrations for Other NP-Complete COPs 1. Introduction 2. The Set Partitioning Problem (SPP) 3. The Vertex Coloring Problem 4. The Multiple Traveling Salesman Problem (mTSP) Chapter 8. Conclusions Bibliography Appendix A. On the (Two) Counter-Example Claims About the Authors Moustapha Diaby is Associate Professor of Production and Operations Management at the University of Connecticut. He received a PhD degree in Management Science/Operations Research, MS degree in Industrial Engineering, and BS degree in Chemical Engineering from University at Buffalo — The State University of New York, USA. His teaching and research interests are in the areas of Mathematical Programming, Manufacturing Systems Modeling and Analysis, Operations and Supply Chain Management, and Project Management. His publications have appeared in European Journal of Operational Research, Information Systems Frontiers Journal, INFORMS Journal on Computing, International Journal of Mathematics in Operational Research, International Journal of Operational Research, International Journal of Production Economics, International Journal of Production Research, International Transactions in Operational Research, Journal of the Operational Research Society, Management Science, Multi-Criteria Decision Analysis, Operations Management Review, Operations Research, and WSEAS Transactions on Mathematics. He serves/has served as a Reviewer and/or ad-hoc Editorial Team Member for many of these, as well as other journals, and for government agencies. Mark H. Karwan is the Praxair Professor in Operations Research at the Department of Industrial and Systems Engineering at University at Buffalo — The State University of New York, USA, where he has taught for 39 years. He has broad expertise in the area of mathematical programming — modeling and algorithmic development. His 31 PhD students have been guided in areas of algorithmic development in integer programming, multiple criteria decision making and ‘mixed’ areas such as integer/nonlinear or integer/multi-criteria. His 100+ publications show diverse application areas such as logistics, production planning under real time pricing, capacitated lot-sizing, hazardous waste routing and security, and military path planning. Techniques to solve these problems come from the fields of linear, nonlinear and integer programming. Funding has come from NSF, ONR and industry. Prof. Karwan’s industry consulting experience has largely been in the industrial gas industry and concerned with all areas of production planning, routing, forecasting and energy use planning and in supporting corporate contracts in military operations research focused on logistics and dynamic resource allocation. He has won multiple teaching awards including the (SUNY) Chancellor’s Award for Excellence in Teaching. His research interests include Discrete Optimization, Multiple Criteria Decision Making, Multilevel Systems, Vehicle Routing and Scheduling, Visual Search, and Industrial Inspection. Preface In this book, we present a generalized framework for formulating hard combinatorial optimization problems (COPs) as polynomial-sized linear programs. Hence, the book offers a new proof of the equality of the computational complexity classes “P” and “NP”. The basic model and its theoretical foundation are developed using the Traveling Salesman Problem (TSP) as an illustration. Then, our proposed generalized framework is presented and illustrated using the TSP also, as well as other well-known hard COPs. The main idea of our approach is to model COPs as flow problems over an assignment- problem (AP) graph. Our variables represent flows over doublets and triplets of arcs of the underlying graph, enabling an inductive path-theoretic argument towards proving that the proposed LP polytope has integral extrema. In the case of the TSP, the doublets and triplets of arcs respectively model doublets and triplets of travel legs, and we show that each extreme point of the resulting LP polytope corresponds to a TSP tour. Although the proposed model draws from the developments in Diaby (2006b; 2007; 2010a; 2010b), the book is fully self-contained, and does not require any familiarity with those previous developments. There are some negative claims that we know of that have been made (through the internet, and in anonymous reviews, respectively) in direct connection to our proposed modeling approach. All of these claims have to do with relaxations of the models in Diaby (2006b; 2007) specifically. These claims are discussed briefly in the introduction chapter, and in complete detail in the appendix. Also, focusing on the TSP, we provide detailed reasons why the existing extended formulations “barriers” (Yannakakis (1991); Fiorini et al. (2011; 2012)) are not applicable to our proposed LP model. Specifically, we show in Chapters 5 and 6 that, in the case of the TSP, our proposed model: (1) is non-symmetric; (2) cannot be extended into a symmetric model using the two-indexed (city-to-city) variables that are traditionally used in defining the standard (i.e., conventional) TSP polytope; (3) does not project to the standard TSP polytope in a well-defined sense; (4) cannot be extended into (and hence, cannot lead to) a polytope which projects to the TSP polytope in a well-defined sense. Although not reported in this book (the focus of which is on theory), our initial empirical testing on hundreds of problems has been consistent with our theoretical developments.

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.