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This updated and revised 2nd edition of the three-volume CombinatorialOptimization series covers a very large set of topics in this area, dealing withfundamental notions and approaches as well as several classical applications ofCombinatorial Optimization.

Combinatorial Optimization is a multidisciplinary field, lying at the interface ofthree major scientific domains: applied mathematics, theoretical computerscience, and management studies. Its focus is on finding the least-cost solutionto a mathematical problem in which each solution is associated with anumerical cost. In many such problems, exhaustive search is not feasible, so theapproach taken is to operate within the domain of optimization problems, inwhich the set of feasible solutions is discrete or can be reduced to discrete, andin which the goal is to find the best solution. Some common problems involvingcombinatorial optimization are the traveling salesman problem and theminimum spanning tree problem.

Combinatorial Optimization is a subset of optimization that is related tooperations research, algorithm theory, and computational complexity theory. Ithas important applications in several fields, including artificial intelligence,mathematics, and software engineering.

This first volume, covering major concepts in the field of CombinatorialOptimization, is divided into three parts:

- Complexity of Combinatorial Optimization Problems, which presents thebasics of worst-case and randomized complexity.

- Classical Solution Methods, which presents the two most well-known methodsfor solving difficult combinatorial optimization problems; Branch and Boundand Dynamic Programming.

- Elements from Mathematical Programming, which presents the fundamentalsfrom mathematical programming-based methods that have been at the heart ofOperations Research since the origins of this field.

The three volumes of this series form a coherent whole. The set of books isintended to be a self-contained treatment requiring only basic understandingand knowledge of a few mathematical theories and concepts. It is intended forresearchers, practitioners and MSc or PhD students.

Vangelis Th. Paschos is Exceptional Class Professor of Computer Science andCombinatorial Optimization at the Paris-Dauphine University and chairman ofthe LAMSADE (Laboratory for the Modeling and the Analysis of Decision AidingSystems) in France.

Concepts ofCombinatorialOptimization2nd Edition Revised and Updated

Edited by Vangelis Th. Paschos

MATHEMATICS AND STATISTICS SERIES

W656-Paschos 1.qxp_Layout 1 01/07/2014 11:24

Concepts of Combinatorial Optimization

Revised and Updated 2nd Edition

Concepts of CombinatorialOptimization

Edited by

Vangelis Th. Paschos

First edition published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons,Inc.© ISTE Ltd 2010Second edition published 2014 in Great Britain and the United States by ISTE Ltd and John Wiley &Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, aspermitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers,or in the case of reprographic reproduction in accordance with the terms and licenses issued by theCLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at theundermentioned address:

ISTE Ltd John Wiley & Sons, Inc.27-37 St George’s Road 111 River StreetLondon SW19 4EU Hoboken, NJ 07030UK USA

www.iste.co.uk www.wiley.com

© ISTE Ltd 2014The rights of Vangelis Th. Paschos to be identified as the author of this work have been asserted by himin accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2014942903

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-84821-656-3

Printed and bound in Great Britain by CPI Group (UK) Ltd., Croydon, Surrey CR0 4YY

Table of Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiVangelis Th. PASCHOS

PART I. COMPLEXITY OFCOMBINATORIALOPTIMIZATION PROBLEMS . . . 1

Chapter 1. Basic Concepts in Algorithms and Complexity Theory . . . . . 3Vangelis Th. PASCHOS

1.1. Algorithmic complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2. Problem complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3. The classes P, NP and NPO . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4. Karp and Turing reductions . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5. NP-completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6. Two examples of NP-complete problems . . . . . . . . . . . . . . . . . . 131.6.1. MIN VERTEX COVER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.6.2. MAX STABLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.7. A few words on strong and weak NP-completeness . . . . . . . . . . . . 161.8. A few other well-known complexity classes . . . . . . . . . . . . . . . . 171.9. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Chapter 2. Randomized Complexity . . . . . . . . . . . . . . . . . . . . . . . . 21Jérémy BARBAY

2.1. Deterministic and probabilistic algorithms . . . . . . . . . . . . . . . . . 222.1.1. Complexity of a Las Vegas algorithm. . . . . . . . . . . . . . . . . . 242.1.2. Probabilistic complexity of a problem. . . . . . . . . . . . . . . . . . 26

2.2. Lower bound technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.1. Definitions and notations . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.2. Minimax theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.3. The Loomis lemma and the Yao principle . . . . . . . . . . . . . . . 33

vi Combinatorial Optimization 1

2.3. Elementary intersection problem . . . . . . . . . . . . . . . . . . . . . . . 352.3.1. Upper bound. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.3.2. Lower bound. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3.3. Probabilistic complexity . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

PART II. CLASSICAL SOLUTIONMETHODS . . . . . . . . . . . . . . . . . . . . . 39

Chapter 3. Branch-and-Bound Methods . . . . . . . . . . . . . . . . . . . . . . 41Irène CHARON and Olivier HUDRY

3.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2. Branch-and-bound method principles . . . . . . . . . . . . . . . . . . . . 433.2.1. Principle of separation . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2.2. Pruning principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2.3. Developing the tree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3. A detailed example: the binary knapsack problem . . . . . . . . . . . . 543.3.1. Calculating the initial bound . . . . . . . . . . . . . . . . . . . . . . . 553.3.2. First principle of separation . . . . . . . . . . . . . . . . . . . . . . . . 573.3.3. Pruning without evaluation . . . . . . . . . . . . . . . . . . . . . . . . 583.3.4. Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.3.5. Complete execution of the branch-and-bound method for findingonly one optimal solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.3.6. First variant: finding all the optimal solutions . . . . . . . . . . . . . 633.3.7. Second variant: best first search strategy . . . . . . . . . . . . . . . . 643.3.8. Third variant: second principle of separation . . . . . . . . . . . . . 65

3.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.5. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Chapter 4. Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . 71Bruno ESCOFFIER and Olivier SPANJAARD

4.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.2. A first example: crossing the bridge . . . . . . . . . . . . . . . . . . . . . 724.3. Formalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3.1. State space, decision set, transition function . . . . . . . . . . . . . . 754.3.2. Feasible policies, comparison relationships and objectives . . . . . 77

4.4. Some other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.4.1. Stock management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.4.2. Shortest path bottleneck in a graph . . . . . . . . . . . . . . . . . . . 814.4.3. Knapsack problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.5. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.5.1. Forward procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Table of Contents vii

4.5.2. Backward procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.5.3. Principles of optimality and monotonicity . . . . . . . . . . . . . . . 86

4.6. Solution of the examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.6.1. Stock management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.6.2. Shortest path bottleneck . . . . . . . . . . . . . . . . . . . . . . . . . . 894.6.3. Knapsack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.7. A few extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.7.1. Partial order and multicriteria optimization . . . . . . . . . . . . . . 914.7.2. Dynamic programming with variables . . . . . . . . . . . . . . . . . 944.7.3. Generalized dynamic programming . . . . . . . . . . . . . . . . . . . 95


PART III. ELEMENTS FROMMATHEMATICAL PROGRAMMING . . . . . . . . . 101

Chapter 5. Mixed Integer Linear Programming Models for CombinatorialOptimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Frédérico DELLA CROCE

5.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.1.1. Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.1.2. The knapsack problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.1.3. The bin-packing problem . . . . . . . . . . . . . . . . . . . . . . . . . 1055.1.4. The set covering/set partitioning problem . . . . . . . . . . . . . . . 1065.1.5. The minimum cost flow problem . . . . . . . . . . . . . . . . . . . . 1075.1.6. The maximum flow problem . . . . . . . . . . . . . . . . . . . . . . . 1085.1.7. The transportation problem . . . . . . . . . . . . . . . . . . . . . . . . 1095.1.8. The assignment problem. . . . . . . . . . . . . . . . . . . . . . . . . . 1105.1.9. The shortest path problem. . . . . . . . . . . . . . . . . . . . . . . . . 111

5.2. General modeling techniques . . . . . . . . . . . . . . . . . . . . . . . . . 1115.2.1. Min-max, max-min, min-abs models . . . . . . . . . . . . . . . . . . 1125.2.2. Handling logic conditions . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.3. More advanced MILP models . . . . . . . . . . . . . . . . . . . . . . . . . 1175.3.1. Location models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.3.2. Graphs and network models . . . . . . . . . . . . . . . . . . . . . . . 1205.3.3. Machine scheduling models. . . . . . . . . . . . . . . . . . . . . . . . 127

5.4. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.5. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Chapter 6. Simplex Algorithms for Linear Programming . . . . . . . . . . . 135Frédérico DELLA CROCE and Andrea GROSSO

6.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.2. Primal and dual programs . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

viii Combinatorial Optimization 1

6.2.1. Optimality conditions and strong duality . . . . . . . . . . . . . . . . 1366.2.2. Symmetry of the duality relation . . . . . . . . . . . . . . . . . . . . . 1376.2.3. Weak duality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386.2.4. Economic interpretation of duality. . . . . . . . . . . . . . . . . . . . 139

6.3. The primal simplex method . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.3.1. Basic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.3.2. Canonical form and reduced costs . . . . . . . . . . . . . . . . . . . . 142

6.4. Bland’s rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1456.4.1. Searching for a feasible solution . . . . . . . . . . . . . . . . . . . . . 146

6.5. Simplex methods for the dual problem . . . . . . . . . . . . . . . . . . . 1476.5.1. The dual simplex method . . . . . . . . . . . . . . . . . . . . . . . . . 1476.5.2. The primal–dual simplex algorithm . . . . . . . . . . . . . . . . . . . 149

6.6. Using reduced costs and pseudo-costs for integer programming . . . . 1526.6.1. Using reduced costs for tightening variable bounds. . . . . . . . . . 1526.6.2. Pseudo-costs for integer programming . . . . . . . . . . . . . . . . . 153

6.7. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Chapter 7. A Survey of some Linear Programming Methods . . . . . . . . . 157Pierre TOLLA

7.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1577.2. Dantzig’s simplex method . . . . . . . . . . . . . . . . . . . . . . . . . . . 1587.2.1. Standard linear programming and the main results . . . . . . . . . . 1587.2.2. Principle of the simplex method . . . . . . . . . . . . . . . . . . . . . 1597.2.3. Putting the problem into canonical form . . . . . . . . . . . . . . . . 1597.2.4. Stopping criterion, heuristics and pivoting . . . . . . . . . . . . . . . 160

7.3. Duality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1627.4. Khachiyan’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1627.5. Interior methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1657.5.1. Karmarkar’s projective algorithm . . . . . . . . . . . . . . . . . . . . 1657.5.2. Primal–dual methods and corrective predictive methods. . . . . . . 1697.5.3. Mehrotra predictor–corrector method . . . . . . . . . . . . . . . . . 181


Chapter 8. Quadratic Optimization in 0–1 Variables . . . . . . . . . . . . . . 189Alain BILLIONNET

8.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1898.2. Pseudo-Boolean functions and set functions . . . . . . . . . . . . . . . . 1908.3. Formalization using pseudo-Boolean functions . . . . . . . . . . . . . . 1918.4. Quadratic pseudo-Boolean functions (qpBf) . . . . . . . . . . . . . . . . 1928.5. Integer optimum and continuous optimum of qpBfs . . . . . . . . . . . 1948.6. Derandomization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Table of Contents ix

8.7. Posiforms and quadratic posiforms . . . . . . . . . . . . . . . . . . . . . . 1968.7.1. Posiform maximization and stability in a graph . . . . . . . . . . . . 1968.7.2. Implication graph associated with a quadratic posiform . . . . . . . 197

8.8. Optimizing a qpBf: special cases and polynomial cases . . . . . . . . . 1988.8.1. Maximizing negative–positive functions . . . . . . . . . . . . . . . . 1988.8.2. Maximizing functions associated with k-trees . . . . . . . . . . . . . 1998.8.3. Maximizing a quadratic posiform whose terms are associatedwith two consecutive arcs of a directed multigraph. . . . . . . . . . . . . . 1998.8.4. Quadratic pseudo-Boolean functions equal to the product of twolinear functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

8.9. Reductions, relaxations, linearizations, bound calculationand persistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2008.9.1. Complementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2008.9.2. Linearization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2018.9.3. Lagrangian duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2028.9.4. Another linearization. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2038.9.5. Convex quadratic relaxation . . . . . . . . . . . . . . . . . . . . . . . 2038.9.6. Positive semi-definite relaxation . . . . . . . . . . . . . . . . . . . . . 2048.9.7. Persistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

8.10. Local optimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2068.11. Exact algorithms and heuristic methods for optimizing qpBfs . . . . . 2088.11.1. Different approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 2088.11.2. An algorithm based on Lagrangian decomposition . . . . . . . . . 2098.11.3. An algorithm based on convex quadratic programming . . . . . . 210

8.12. Approximation algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 2118.12.1. A 2-approximation algorithm for maximizing a quadraticposiform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2118.12.2. MAX-SAT approximation . . . . . . . . . . . . . . . . . . . . . . . . 213

8.13. Optimizing a quadratic pseudo-Boolean function with linearconstraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2138.13.1. Examples of formulations . . . . . . . . . . . . . . . . . . . . . . . . 2148.13.2. Some polynomial and pseudo-polynomial cases . . . . . . . . . . 2178.13.3. Complementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

8.14. Linearization, convexification and Lagrangian relaxation foroptimizing a qpBf with linear constraints . . . . . . . . . . . . . . . . . . . . . 2208.14.1. Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2218.14.2. Convexification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2228.14.3. Lagrangian duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

8.15. ε-Approximation algorithms for optimizing a qpBf with linearconstraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2238.16. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

x Combinatorial Optimization 1

Chapter 9. Column Generation in Integer Linear Programming . . . . . . 235Irène LOISEAU, Alberto CESELLI, Nelson MACULAN and Matteo SALANI

9.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2359.2. A column generation method for a bounded variable linearprogramming problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2369.3. An inequality to eliminate the generation of a 0–1 column . . . . . . . . 2389.4. Formulations for an integer linear program . . . . . . . . . . . . . . . . . 2409.5. Solving an integer linear program using column generation . . . . . . . 2439.5.1. Auxiliary problem (pricing problem) . . . . . . . . . . . . . . . . . . 2439.5.2. Branching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

9.6. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2479.6.1. The p-medians problem . . . . . . . . . . . . . . . . . . . . . . . . . . 2479.6.2. Vehicle routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

9.7. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

Chapter 10. Polyhedral Approaches. . . . . . . . . . . . . . . . . . . . . . . . . 261Ali Ridha MAHJOUB

10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26110.2. Polyhedra, faces and facets . . . . . . . . . . . . . . . . . . . . . . . . . . 26510.2.1. Polyhedra, polytopes and dimension . . . . . . . . . . . . . . . . . . 26510.2.2. Faces and facets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

10.3. Combinatorial optimization and linear programming . . . . . . . . . . 27610.3.1. Associated polytope . . . . . . . . . . . . . . . . . . . . . . . . . . . 27610.3.2. Extreme points and extreme rays . . . . . . . . . . . . . . . . . . . . 279

10.4. Proof techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28210.4.1. Facet proof techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 28310.4.2. Integrality techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . 287

10.5. Integer polyhedra and min–max relations . . . . . . . . . . . . . . . . . 29310.5.1. Duality and combinatorial optimization . . . . . . . . . . . . . . . . 29310.5.2. Totally unimodular matrices. . . . . . . . . . . . . . . . . . . . . . . 29410.5.3. Totally dual integral systems . . . . . . . . . . . . . . . . . . . . . . 29610.5.4. Blocking and antiblocking polyhedral . . . . . . . . . . . . . . . . . 297

10.6. Cutting-plane method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30110.6.1. The Chvátal–Gomory method. . . . . . . . . . . . . . . . . . . . . . 30210.6.2. Cutting-plane algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 30410.6.3. Branch-and-cut algorithms. . . . . . . . . . . . . . . . . . . . . . . . 30510.6.4. Separation and optimization . . . . . . . . . . . . . . . . . . . . . . . 306

10.7. The maximum cut problem . . . . . . . . . . . . . . . . . . . . . . . . . 30810.7.1. Spin glass models and the maximum cut problem . . . . . . . . . 30910.7.2. The cut polytope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

10.8. The survivable network design problem . . . . . . . . . . . . . . . . . . 31310.8.1. Formulation and associated polyhedron . . . . . . . . . . . . . . . . 314

Table of Contents xi

10.8.2. Valid inequalities and separation . . . . . . . . . . . . . . . . . . . 31510.8.3. A branch-and-cut algorithm . . . . . . . . . . . . . . . . . . . . . . . 318

10.9. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31910.10. Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

Chapter 11. Constraint Programming . . . . . . . . . . . . . . . . . . . . . . . 325Claude LE PAPE

11.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32511.2. Problem definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32711.3. Decision operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32811.4. Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33011.5. Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33311.5.1. Branching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33311.5.2. Exploration strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . 335

11.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33611.7. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

General Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

List of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

Summary of Other Volumes in the Series . . . . . . . . . . . . . . . . . . . . . 371

Preface

What is combinatorial optimization? There are, in my opinion, as manydefinitions as there are researchers in this domain, each one as valid as the other. Forme, it is above all the art of understanding a real, natural problem, and being able totransform it into a mathematical model. It is the art of studying this model in orderto extract its structural properties and the characteristics of the solutions of themodeled problem. It is the art of exploiting these characteristics in order todetermine algorithms that calculate the solutions but also to show the limits ineconomy and efficiency of these algorithms. Lastly, it is the art of enriching orabstracting existing models in order to increase their strength, portability, and abilityto describe mathematically (and computationally) other problems, which may ormay not be similar to the problems that inspired the initial models.

Seen in this light, we can easily understand why combinatorial optimization is atthe heart of the junction of scientific disciplines as rich and different as theoreticalcomputer science, pure and applied, discrete and continuous mathematics,mathematical economics, and quantitative management. It is inspired by these, andenriches them all.

This revised and updated second edition of Concepts of CombinatorialOptimization, is the first volume in a series of books entitled CombinatorialOptimization. It tries, along with the other volumes in the set, to embody the idea ofcombinatorial optimization. The subjects of this volume cover themes that areconsidered to constitute the hard core of combinatorial optimization. The book isdivided into three parts:

– Part I: Complexity of Combinatorial Optimization Problems;

– Part II: Classical Solution Methods;

– Part III: Elements from Mathematical Programming.

xiv Combinatorial Optimization 1

In the first part, Chapter 1 introduces the fundamentals of the theory of(deterministic) complexity and of algorithm analysis. In Chapter 2, the contextchanges and we consider algorithms that make decisions by “tossing a coin”. Ateach stage of the resolution of a problem, several alternatives have to be considered,each one occurring with a certain probability. This is the context of probabilistic (orrandomized) algorithms, which is described in this chapter.

In the second part some methods are introduced that make up the great classicsof combinatorial optimization: branch-and-bound and dynamic programming. Theformer is perhaps the most well known and the most popular when we try to find anoptimal solution to a difficult combinatorial optimization problem. Chapter 3 gives athorough overview of this method as well as of some of the most well-known treesearch methods based upon branch-and-bound. What can we say about dynamicprogramming, presented in Chapter 4? It has considerable reach and scope, and verymany optimization problems have optimal solution algorithms that use it as theircentral method.

The third part is centered around mathematical programming, considered to bethe heart of combinatorial optimization and operational research. In Chapter 5, alarge number of linear models and an equally large number of combinatorialoptimization problems are set out and discussed. In Chapter 6, the main simplexalgorithms for linear programming, such as the primal simplex algorithm, the dualsimplex algorithm, and the primal–dual simplex algorithm are introduced. Chapter 7introduces some classical linear programming methods, while Chapter 8 introducesquadratic integer optimization methods. Chapter 9 describes a series of resolutionmethods currently widely in use, namely column generation. Chapter 10 focuses onpolyhedral methods, almost 60 years old but still relevant to combinatorialoptimization research. Lastly, Chapter 11 introduces a more contemporary, butextremely interesting, subject, namely constraint programming.

This book is intended for novice researchers, or even Master’s students, as muchas for senior researchers. Master’s students will probably need a little basicknowledge of graph theory and mathematical (especially linear) programming to beable to read the book comfortably, even though the authors have been careful to givedefinitions of all the concepts they use in their chapters. In any case, to improvetheir knowledge of graph theory, readers are invited to consult a great, flagship bookfrom one of our gurus, Claude Berge: Graphs and Hypergraphs, North Holland,1973. For linear programming, there is a multitude of good books that the readercould consult, for example V. Chvátal, Linear Programming, W.H. Freeman, 1983,or M. Minoux, Programmation mathématique: théorie et algorithmes, Dunod, 1983.

In this revised second edition I would like to thank again the authors who,despite their many responsibilities and commitments (which is the lot of any

Preface xv

university academic), agreed to participate in the book by writing chapters in theirareas of expertise and, at the same time, to take part in a very tricky exercise: writingchapters that are both educational and high-level science at the same time.

The second edition of this book could never have come into being without theoriginal proposal of Jean-Charles Pomerol, President of the scientific board at ISTE,and Sami Ménascé and Raphaël Ménascé, the President and Vice-President of ISTE,respectively. I once again give my warmest thanks to them for their insistence andencouragement.

Vangelis Th. PASCHOSJune 2014

PART I

Complexity of CombinatorialOptimization Problems

Chapter 1

Basic Concepts in Algorithmsand Complexity Theory

1.1. Algorithmic complexity

In algorithmic theory, a problem is a general question to which we wish to find ananswer. This question usually has parameters or variables the values of which haveyet to be determined. A problem is posed by giving a list of these parameters as wellas the properties to which the answer must conform. An instance of a problem isobtained by giving explicit values to each of the parameters of the instanced problem.

An algorithm is a sequence of elementary operations (variable affectation, tests,forks, etc.) that, when given an instance of a problem as input, gives the solution ofthis problem as output after execution of the final operation.

The two most important parameters for measuring the quality of an algorithm are:its execution time and the memory space that it uses. The first parameter is expressedin terms of the number of instructions necessary to run the algorithm. The use of thenumber of instructions as a unit of time is justified by the fact that the same programwill use the same number of instructions on two different machines but the time takenwill vary, depending on the respective speeds of the machines. We generally considerthat an instruction equates to an elementary operation, for example an assignment, atest, an addition, a multiplication, a trace, etc. What we call the complexity in time orsimply the complexity of an algorithm gives us an indication of the time it will take tosolve a problem of a given size. In reality this is a function that associates an order of

Chapter written by Vangelis Th. PASCHOS.

4 Combinatorial Optimization 1

magnitude1 of the number of instructions necessary for the solution of a given problemwith the size of an instance of that problem. The second parameter corresponds to thenumber of memory units used by the algorithm to solve a problem. The complexityin space is a function that associates an order of magnitude of the number of memoryunits used for the operations necessary for the solution of a given problem with thesize of an instance of that problem.

There are several sets of hypotheses concerning the “standard configuration” thatwe use as a basis for measuring the complexity of an algorithm. The most commonlyused framework is the one known as “worst-case”. Here, the complexity of an algo-rithm is the number of operations carried out on the instance that represents the worstconfiguration, amongst those of a fixed size, for its execution; this is the frameworkused in most of this book. However, it is not the only framework for analyzing thecomplexity of an algorithm. Another framework often used is “average analysis”.This kind of analysis consists of finding, for a fixed size (of the instance) n, the aver-age execution time of an algorithm on all the instances of size n; we assume that forthis analysis the probability of each instance occurring follows a specific distributionpattern. More often than not, this distribution pattern is considered to be uniform.There are three main reasons for the worst-case analysis being used more often thanthe average analysis. The first is psychological: the worst-case result tells us for cer-tain that the algorithm being analyzed can never have a level of complexity higherthan that shown by this analysis; in other words, the result we have obtained givesus an upper bound on the complexity of our algorithm. The second reason is mathe-matical: results from a worst-case analysis are often easier to obtain than those froman average analysis, which very often requires mathematical tools and more complexanalysis. The third reason is “analysis portability”: the validity of an average analysisis limited by the assumptions made about the distribution pattern of the instances; ifthe assumptions change, then the original analysis is no longer valid.

1.2. Problem complexity

The definition of the complexity of an algorithm can be easily transposed to prob-lems. Informally, the complexity of a problem is equal to the complexity of the bestalgorithm that solves it (this definition is valid independently of which framework weuse).

Let us take a size n and a function f(n). Thus:

– TIMEf(n) is the class of problems for which the complexity (in time) of aninstance of size n is in O(f(n)).

1. Orders of magnitude are defined as follows: given two functions f and g: f = O(g) if andonly if ∃(k, k ) ∈ (R+, R) such that f kg +k ; f = o(g) if and only if limn→∞(f/g) = 0;f = Ω(g) if and only if g = o(f); f = Θ(g) if and only if f = O(g) and g = O(f).

Basic Concepts in Algorithms and Complexity Theory 5

– SPACEf(n) is the class of problems that can be solved, for an instance of size n,by using a memory space of O(f(n)).

We can now specify the following general classes of complexity:

– P is the class of all the problems that can be solved in a time that is a polynomialfunction of their instance size, that is P = ∪∞

k= 0 TIMEnk.

– EXPTIME is the class of problems that can be solved in a time that is an expo-nential function of their instance size, that is EXPTIME = ∪∞

k= 0 TIME2nk

.

– PSPACE is the class of all the problems that can be solved using a mem-ory space that is a polynomial function of their instance size, that is PSPACE =∪∞

k= 0 SPACEnk.

With respect to the classes that we have just defined, we have the following re-lations: P ⊆ PSPACE ⊆ EXPTIME and P ⊂ EXPTIME. Knowing whether theinclusions of the first relation are strict or not is still an open problem.

Almost all combinatorial optimization problems can be classified, from an algo-rithmic complexity point of view, into two large categories. Polynomial problemscan be solved optimally by algorithms of polynomial complexity, that is in O(nk),where k is a constant independent of n (this is the class P that we have already de-fined). Non-polynomial problems are those for which the best algorithms (those givingan optimum solution) are of a “super-polynomial” complexity, that is in O(f(n)g( n) ),where f and g are increasing functions in n and limn→∞ g(n) = ∞. All these prob-lems contain the class EXPTIME.

The definition of any algorithmic problem (and even more so in the case of anycombinatorial optimization problem) comprises two parts. The first gives the instanceof the problem, that is the type of its variables (a graph, a set, a logical expression,etc.). The second part gives the type and properties to which the expected solutionmust conform. In the complexity theory case, algorithmic problems can be classifiedinto three categories:

– decision problems;

– optimum value calculation problems;

– optimization problems.

Decision problems are questions concerning the existence, for a given instance, ofa configuration such that this configuration itself, or its value, conforms to certainproperties. The solution to a decision problem is an answer to the question associatedwith the problem. In other words, this solution can be:

– either “yes, such a configuration does exist”;

– or “no, such a configuration does not exist”.


Let us consider as an example the conjunctive normal form satisfiability problem,known in the literature as SAT: “Given a set U of n Boolean variables x1 , . . . , xn and aset C of m clauses2 C1 , . . . , Cm, is there a model for the expression φ = C1 ∧. . .∧Cm;i.e. is there an assignment of the values 0 or 1 to the variables such that φ = 1?”.For an instance φ of this problem, if φ allows a model then the solution (the correctanswer) is yes, otherwise the solution is no.

Let us now consider the MIN TSP problem, defined as follows: given a completegraph Kn over n vertices for which each edge e ∈ E(Kn) has a value d(e) > 0, weare looking for a Hamiltonian cycle H ⊂ E (a partial closely related graph such thateach vertex is of 2 degrees) that minimizes the quantity e∈H d(e). Let us assumethat for this problem we have, as well as the complete graph Kn and the vector d,costs on the edges Kn of a constant K and that we are looking not to determine thesmallest (in terms of total cost) Hamiltonian cycle, but rather to answer the followingquestion: “Does there exist a Hamiltonian cycle of total distance less than or equalto K?”. Here, once more, the solution is either yes if such a cycle exists, or no if itdoes not.

For optimum value calculation problems, we are looking to calculate the value ofthe optimum solution (and not the solution itself).

In the case of the MIN TSP for example, the optimum associated value calculationproblem comes down to calculating the cost of the smallest Hamiltonian cycle, andnot the cycle itself.

Optimization problems, which are naturally of interest to us in this book, are thosefor which we are looking to establish the best solution amongst those satisfying certainproperties given by the very definition of the problem. An optimization problem maybe seen as a mathematical program of the form:

opt v (x)x ∈ CI

where x is the vector describing the solution3, v(x) is the objective function, CI is theproblem’s constraint set, set out for the instance I (in other words, CI sets out both theinstance and the properties of the solution that we are looking to find for this instance),and opt ∈ max, min. An optimum solution of I is a vector x∗ ∈ argoptv(x) :x ∈ CI. The quantity v(x∗) is known as the objective value or value of the problem.A solution x ∈ CI is known as a feasible solution.

2. We associate two literals x and x with a Boolean variable x, that is the variable itself and itsnegation; a clause is a disjunction of the literals.3. For the combinatorial optimization problems that concern us, we can assume that the com-ponents of this vector are 0 or 1 or, if need be, integers.


Let us consider the problem MIN WEIGHTED VERTEX COVER4. An instance of thisproblem (given by the information from the incident matrix A, of dimension m × n,of a graph G(V, E) of order n with |E| = m and a vector w, of dimension n of thecosts of the edges of V ), can be expressed in terms of a linear program in integers asfollows: ⎧⎨⎩

min w · xA.x 1x ∈ 0, 1n

where x is a vector from 0, 1 of dimension n such that xi = 1 if the vertex vi ∈ Vis included in the solution, xi = 0 if it is not included. The block of m constraintsA.x 1 expresses the fact that for each edge at least one of these extremes must beincluded in the solution. The feasible solutions are all the transversals of G and theoptimum solution is a transversal of G of minimum total weight, that is a transversalcorresponding to a feasible vector consisting of a maximum number of 1.

The solution to an optimization problem includes an evaluation of the optimumvalue. Therefore, an optimum value calculation problem can be associated with anoptimization problem. Moreover, optimization problems always have a decisionalvariant as shown in the MIN TSP example above.

1.3. The classes P, NP and NPO

Let us consider a decision problem Π. If for any instance I of Π a solution (thatis a correct answer to the question that states Π) of I can be found algorithmically inpolynomial time, that is in O(|I|k) stages, where |I| is the size of I , then Π is calleda polynomial problem and the algorithm that solves it a polynomial algorithm (let usremember that polynomial problems make up the P class).

For reasons of simplicity, we will assume in what follows that the solution to adecision problem is:

– either “yes, such a solution exists, and this is it”;

– or “no, such a solution does not exist”.

In other words, if, to solve a problem, we could consult an “oracle”, it wouldprovide us with an answer of not just a yes or no but also, in the first case, a certificate

4. Given a graph G(V, E) of order n, in the MIN VERTEX COVER problem we are looking tofind a smallest transversal of G, that is a set V ⊆ V such that for every edge (u, v) ∈ E,either u ∈ V , or v ∈ V of minimum size; we denote by MIN WEIGHTED VERTEX COVER theversion of MIN VERTEX COVER where a positive weight is associated with each vertex and theobjective is to find a transversal of G that minimizes the sum of the vertex weights.


proving the veracity of the yes. This testimony is simply a solution proposal that theoracle “asserts” as being the real solution to our problem.

Let us consider the decision problems for which the validity of the certificate canbe verified in polynomial time. These problems form the class NP.

DEFINITION 1.1.– A decision problem Π is in NP if the validity of all solutions of Πis verifiable in polynomial time.

For example, the SAT problem belongs to NP. Indeed, given the assignment of thevalues 0, 1 to the variables of an instance φ of this problem, we can, with at most nmapplications of the connector ∨, decide whether the proposed assignment is a modelfor φ, that is whether it satisfies all the clauses.

Therefore, we can easily see that the decisional variant of MIN TSP seen previouslyalso belongs to NP.

Definition 1.1 can be extended to optimization problems. Let us consider an opti-mization problem Π and let us assume that each instance I of Π conforms to the threefollowing properties:

1) The feasibility of a solution can be verified in polynomial time.

2) The value of a feasible solution can be calculated in polynomial time.

3) There is at least one feasible solution that can be calculated in polynomial time.

Thus, Π belongs to the class NPO. In other words, the class NPO is the class ofoptimization problems for which the decisional variant is in NP. We can thereforedefine the class PO of optimization problems for which the decisional variant belongsto the class P. In other words, PO is the class of problems that can be optimally solvedin polynomial time.

We note that the class NP has been defined (see definition 1.1) without explicitreference to the optimum solution of its problems, but by reference to the verificationof a given solution. Evidently, the condition of belonging to P being stronger thanthat of belonging to NP (what can be solved can be verified), we have the obviousinclusion of : P ⊆ NP (Figure 1.1).

“What is the complexity of the problems in NP \ P?”. The best general result onthe complexity of the solution of problems from NP is as follows [GAR 79].

THEOREM 1.1.– For any problem Π ∈ NP, there is a polynomial pΠ such that eachinstance I of Π can be solved by an algorithm of complexity O(2pΠ ( |I|) ).

In fact, theorem 1.1 merely gives an upper limit on the complexity of problemsinNP, but no lower limit. The diagram in Figure 1.1 is nothing more than a conjecture,


NP

P

Figure 1.1. P and NP (under the assumption P = NP)

and although almost all researchers in complexity are completely convinced of itsveracity, it has still not been proved. The question “Is P equal to or different fromNP?” is the biggest open question in computer science and one of the best known inmathematics.

1.4. Karp and Turing reductions

As we have seen, problems in NP \ P are considered to be algorithmically moredifficult than problems in P. A large number of problems in NP \ P are very stronglybound to each other through the concept of polynomial reduction.

The principle of reducing a problem Π to a problem Π consists of considering theproblem Π as a specific case of Π , modulo a slight transformation. If this transfor-mation is polynomial, and we know that we can solve Π in polynomial time, we willalso be able to solve Π in polynomial time. Reduction is thus a means of transferringthe result of solving one problem to another; in the same way it is a tool for classifyingproblems according to the level of difficulty of their solution.

We will start with the classic Karp reduction (for the classNP) [GAR 79, KAR 72].This links two decision problems by the possibility of their optimum (and simultane-ous) solution in polynomial time. In the following, given a problem Π, let IΠ be all ofits instances (we assume that each instance I ∈ IΠ is identifiable in polynomial timein |I|). Let OΠ be the subset of IΠ for which the solution is yes; OΠ is also known asthe set of yes-instances (or positive instances) of Π.

DEFINITION 1.2.– Given two decision problems Π1 and Π2 , a Karp reduction (orpolynomial transformation) is a function f : IΠ 1 → IΠ 2 , which can be calculated inpolynomial time, such that, given a solution for f(I), we are able to find a solutionfor I in polynomial time in |I| (the size of the instance I).


A Karp reduction of a decision problem Π1 to a decision problem Π2 implies theexistence of an algorithm A1 for Π1 that uses an algorithm A2 for Π2 . Given anyinstance I1 ∈ IΠ 1 , the algorithm A1 constructs an instance I2 ∈ IΠ 2 ; it executes thealgorithm A2, which calculates a solution on I2 , then A1 transforms this solution intoa solution for Π1 on I1 . If A2 is polynomial, then A1 is also polynomial.

Following on from this, we can state another reduction, known in the literatureas the Turing reduction, which is better adapted to optimization problems. In whatfollows, we define a problem Π as a couple (IΠ , SolΠ ), where SolΠ is the set ofsolutions for Π (we denote by SolΠ (I) the set of solutions for the instance I ∈ IΠ ).

DEFINITION 1.3.– A Turing reduction of a problem Π1 to a problem Π2 is an algo-rithm A1 that solves Π1 by using (possibly several times) an algorithm A2 for Π2 insuch a way that if A2 is polynomial, then A1 is also polynomial.

The Karp and Turing reductions are transitive: if Π1 is reduced (by one of thesetwo reductions) to Π2 and Π2 is reduced to Π3 , then Π1 reduces to Π3 . We cantherefore see that both reductions preserve membership of the class P in the sense thatif Π reduces to Π and Π ∈ P, then Π ∈ P.

For more details on both the Karp and Turing reductions refer to [AUS 99, GAR 79,PAS 04]. In Garey and Johnson [GAR 79] (Chapter 5) there is also a very interestinghistorical summary of the development of the ideas and terms that have led to thestructure of complexity theory as we know it today.

1.5. NP-completeness

From the definition of the two reductions in the preceding section, if Π reducesto Π, then Π can reasonably be considered as at least as difficult as Π (regardingtheir solution by polynomial algorithms), in the sense that a polynomial algorithmfor Π would have sufficed to solve not only Π itself, but equally Π . Let us confineourselves to the Karp reduction. By using it, we can highlight a problem Π∗ ∈ NPsuch that any other problem Π ∈ NP reduces to Π∗ [COO 71, GAR 79, FAG 74].Such a problem is, as we have mentioned, the most difficult problem of the class NP.Therefore we can show [GAR 79, KAR 72] that there are other problems Π∗ ∈ NPsuch that Π∗ reduces to Π∗ . In this way we expose a family of problems such thatany problem in NP reduces (remembering that the Karp reduction is transitive) to oneof its problems. This family has, of course, the following properties:

– It is made up of the most difficult problems of NP.

– A polynomial algorithm for at least one of its problems would have been suffi-cient to solve, in polynomial time, all the other problems of this family (and indeedany problem in NP).


The problems from this family are NP-complete problems and the class of theseproblems is called the NP-complete class.

DEFINITION 1.4.– A decision problem Π is NP-complete if, and only if, it fulfills thefollowing two conditions:

1) Π ∈ NP ;

2) ∀Π ∈ NP, Π reduces to Π by a Karp reduction.

Of course, a notion of NP-completeness very similar to that of definition 1.4 canalso be based on the Turing reduction.

The following application of definition 1.3 is very often used to show the NP-completeness of a problem. Let Π1 = (IΠ 1 , SolΠ 1) and Π2 = (IΠ 2 , SolΠ 2) be twoproblems, and let (f, g) be a pair of functions, which can be calculated in polynomialtime, where:

– f : IΠ 1 → IΠ 2 is such that for any instance I ∈ IΠ 1 , f(I) ∈ IΠ 2 ;

– g : IΠ 1 × SolΠ 2 → SolΠ 1 is such that for every pair (I, S) ∈ (IΠ 1 ×SolΠ 2(f(I))), g(I, S) ∈ SolΠ 1(I).

Let us assume that there is a polynomial algorithm A for the problem Π2 . In this case,the algorithm f A g is a (polynomial) Turing reduction.

A problem that fulfills condition 2 of definition 1.4 (without necessarily checkingcondition 1) is called NP-hard5. It follows that a decision problem Π is NP-completeif and only if it belongs to NP and it is NP-hard.

With the class NP-complete, we can further refine (Figure 1.2) the world of NP.Of course, if P = NP, the three classes from Figure 1.2 coincide; moreover, under theassumption P = NP, the classes P and NP-complete do not intersect.

Let us denote by NP-intermediate the class NP\(P∪NP−complete). Informally,this concerns the class of problems of intermediate difficulty, that is problems that aremore difficult than those from P but easier than those from NP-complete. More for-mally, for two complexity classes C and C such that C ⊆ C, and a reduction R pre-serving the membership of C , a problem is C-intermediate if it is neither C-completeunder R, nor belongs to C . Under the Karp reduction, the class NP-intermediate isnot empty [LAD 75].

Let us note that the idea of NP-completeness goes hand in hand with decisionproblems. When dealing with optimization problems, the appropriate term, used in

5. These are the starred problems in the appendices of [GAR 79].


NP

P

NP-complete

NP-intermediate

Figure 1.2. P, NP and NP-complete (under the assumption P = NP)

the literature, is NP-hard6. A problem of NPO is NP-hard if and only if its decisionalvariant is an NP-complete problem.

The problem SAT was the first problem shown to be NP-complete (the proof ofthis important result can be found in [COO 71]). The reduction used (often calledgeneric reduction) is based on the theory of recursive languages and Turing ma-chines (see [HOP 79, LEW 81] for more details and depth on the Turing machineconcept; also, language-problem correspondence is very well described in [GAR 79,LEW 81]). The general idea of generic reduction, also often called the “Cook–Levintechnique (or theory)”, is as follows: For a generic decision (language) problembelonging to NP, we describe, using a normal conjunctive form, the working of anon-deterministic algorithm (Turing machine) that solves (decides) this problem (lan-guage).

The second problem shown to beNP-complete [GAR 79, KAR 72] was the variantof SAT, written 3SAT, where no clause contains more than three literals. The reductionhere is from SAT [GAR 79, PAP 81]. More generally, for all k 3, the kSAT problems(that is the problems defined on normal conjunctive forms where each clause containsno more than k literals) are all NP-complete.

It must be noted that the problem 2SAT, where all normal conjunctive form clausescontain at most two literals, is polynomial [EVE 76]. It should also be noted thatin [KAR 72], where there is a list of the first 21 NP-complete problems, the problemof linear programming in real numbers was mentioned as a probable problem from the

6. There is a clash with this term when it is used for optimization problems and when it is usedin the sense of property 2 of definition 1.4, where it means that a decision problem Π is harderthan any other decision problem Π ∈ NP.

conceptsofcombinatorialoptimization...concepts of combinatorial optimization 2 nd edition revised...

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