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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
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A catalogue record for this book is available from the British Library.
Copyright © 2016 by World Scientific Publishing Co. Pte. Ltd.
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“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.
