Approximation and Complexity in Numerical Optimization: Continuous and Discrete Problems

597 Pages·2000·17.87 MB·English

by  James Abello

Approximation and Complexity in Numerical Optimization
Nonconvex Optimization and Its Applications
Volume 42
Managing Editor:
Panos Pardalos
University ofF lorida, U.S.A.
Advisory Board:
J.R. Birge
University ofM ichigan, U.S.A.
Ding-ZhuDu
University ofM innesota, U.S.A.
C. A. Floudas
Princeton University, U.S.A.
J.Mockus
Lithuanian Academy of Sciences, Lithuania
H. D. Sherali
Virginia Polytechnic Institute and State University, U.S.A.
G. Stavroulakis
Technical University Braunschweig, Germany
The titles published in this series are listed at the end o/this volume.
Approximation and Complexity
in Numerical Optimization
Continuous and Discrete Problems
Edited by
Panos M. Pardalos
Center for Applied Optimization,
Department of Industrial and Systems Engineering,
University of Florida, U.S.A.
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A c.l.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4419-4829-8 ISBN 978-1-4757-3145-3 (eBook)
DOI 10.1007/978-1-4757-3145-3
Printed an acid-free paper
AII Rights Reserved
© 2000 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2000
Softcover reprint ofthe hardcover lst edition 2000
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
incIuding photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Contents
Preface ......................................................................... xv
Navigating Graph Surfaces .................................................. 1
J. Abello, S. Krishnan
1. Introduction ................................................................. 2
2. What is a hierarchical graph surface? ......................................... 2
3. Problem statement ........................................................... 3
4. Navigating ................................................................... 4
5. From multi-digraphs to surfaces .............................................. 7
6. Triangulation and simplification algorithm .................................... 9
7. Visual navigation of graph surfaces .......................................... 11
8. Applications and open questions ............................................. 13
9. Conclusions ................................................................. 14
References ...................................................................... 15
The Steiner Ratio of Lp-planes ............................................. 17
J. Albrecht, D. Cieslik
1. Introduction ................................................................. 18
2. Well - known bounds and exact values ....................................... 20
3. Better upper bounds ......................................................... 22
4. Sets with four elements ...................................................... 23
5. The Steiner ratio of dual planes .............................................. 26
6. Concluding remarks .......................................................... 27
References ...................................................................... 29
Hamiltonian Cycle Problem via Markov Chains
and Min-type Approaches .................................................. 31
M. Andramonov, J. Filar, P. Pardalos, A. Rubinov
1. Introduction ................................................................. 32
2. Formulation ................................................................. 32
3. Numerical experiments ....................................................... 36
4. Appendix 1: An embedding of the Hep in a Markov decision process ........ 38
5. Appendix 2: Global minimization via min type functions ..................... 42
References ...................................................................... 44
vi
Solving Large Scale Uncapacitated Facility
Location Problems ........................................................... 48
F. Barahona, F. Chudak
1. Introduction ................................................................. 48
2. Solving the linear programming relaxation ................................... 50
3. Randomized rounding ........................................................ 53
4. A new heuristic for the UFLP ................................................ 54
5. Computational experiments .................................................. 54
References ...................................................................... 61
A Branch - and - Bound Procedure for the
Largest Clique in a Graph .................................................. 63
E. R. Barnes
1. Introduction ................................................................. 63
2. Upper and lower bounds ..................................................... 64
3. Sharper bounds .............................................................. 69
4. The branch - and - bound procedure ......................................... 73
References ...................................................................... 76
A New "Annealed" Heuristic for the
Maximum Clique Problem .................................................. 78
1. M. Bomze, M. Budinich, M. Pelillo and C. Rossi
1. Introduction ................................................................. 79
2. Evolution towards the maximum clique and the
annealing parameter ............................................................ 79
3. A prototypical example ...................................................... 84
4. The annealed replication heuristic ............................................ 86
5. Experimental results ......................................................... 88
6. Conclusions .................................................................. 92
References ...................................................................... 93
Inapproximability of some
Geometric and Quadratic Optimization Problems ....................... 96
A. Brieden, P. Gritzmann, V. Klee
1. Introduction ................................................................ 97
2. Definitions and statements of main results ................................... 98
3. Upper bounds .............................................................. 100
4. From logic to geometry ..................................................... 102
vii
5. From [-1, l]-PARMAX to [O,l]-PARMAX ................................... 107
6. From [0, l]-PARMAX to SIMPLEX-WIDTH ................................... 109
References ..................................................................... 114
Convergence Rate of the P-Algorithm for
Optimization of Continious Functions .................................... 116
J. M. Calvin
1. Introduction ................................................................ 116
2. Notation and background ................................................... 119
3. Point process of observations ...................................... ~ ......... 120
4. Asymptotic normalized error ................................................ 126
References ..................................................................... 128
Application of Semidefinite Programming to
Circuit Partitioning ........................................................ 130
C. C. Choi, Y. Ye
1. Introduction ................................................................ 130
2. Translating hypergraph ....................................•................ 132
3. SDP method ................................................................ 133
4. Rank reduction ............................................................. 134
5. Test result .................................................................. 135
6. Final remarks ................................................................ 136
References ..................................................................... 136
Combinatorial Problems Arising in Deregulated
Electrical Power Industry: Survey and Future Directions .............. 138
D. Cook, G. Hicks, V. Faber, M. V. Marathe,
A. Srinivasan, Y. J. Sussmann, H. Thornquist
1. Introduction: the changing face of power industry ........................... 139
2. Problem formulations ....................................................... 142
3. Formulating electric flow problems .......................................... 143
4. Preliminary definitions ............................................ " ., ...... 143
5. Related work ............................................................... 148
6. Hardness results ............................................................ 149
7. Easiness results: exact and approximation algorithms ....................... 150
8. Experimental analysis ....................................................... 151
9. Current work ............................................................... 154
10. Conclusions ................................................................ 154
References ..................................................................... 157
viii
On Approximating a Scheduling Problem ............................... 163
P. Crescenzi, X. Deng, C. H. Papadimitriou
1. Introduction ................................................................ 164
2. Inapproximability ........................................................... 166
3. Approximation algorithms .................................................. 168
References ..................................................................... 173
Models and Solution for On-Demand Data
Delivery Problems .......................................................... 175
M. C. Ferris, R. R.Meyer
1. Introduction ................................................................ 176
2. Skyscraper delivery techniques .............................................. 177
3. MIP model and solution .................................................... 180
4. Conclusion .................................................................. 186
References ..................................................................... 187
Complexity and Experimental Evaluation of
Primal-Dual Shortest Path Tree Algorithms ............................ 189
P. Festa, R. Cerulli, G. Raiconi
1. Mathematical model ........................................................ 190
2. A generic shortest path algorithm ........ '" ................................ 191
3. Implementations of the generic algorithm ................................... 192
4. Auction algorithms ......................................................... 195
5. Computational results ...................................................... 201
6. Conclusions ................................................................. 205
7. Appendix ................................................................... 205
Acknowledgement ............................................................ 207
References ..................................................................... 207
Machine Partitioning and Scheduling under
Fault-Tolerance Constraints ............................................... 209
D. A. Fotakis, P. G. Spirakis
1. Introduction ................................................................ 210
2. Preliminaries ............................................................... 212
3. Fault-tolerant partition of identical speeds .................................. 217
4. Fault-tolerant partition of related speeds .................................... 221
5. Assignments on identical speed machines .................................... 234
6. Assignments on related speed machines ..................................... 240
ix
7. Open problems ............................................................. 242
References ..................................................................... 243
Finding Optimal Boolean Classifiers ..................................... 245
J. Franco
1. Introduction ................................................................ 246
2. Error analysis ............................................................... 246
3. An example ................................................................. 251
4. Models of data generation and the structure of data ......................... 254
5. What's next? ............................................................... 262
Acknowledgements ........................................................... 263
Appendices .................................................................... 264
A. Details of experiments with k - convex closures ............................. 264
B. Examples of experiments with vine models .................................. 268
C. Program descriptions ....................................................... 274
D. Variograms for truth circles ................................................. 285
E. Variograms for truth sub-hypercubes ........................................ 286
References ..................................................................... 286
Tighter Bounds on the Performance of First Fit
Bin Packing ................................................................. 287
M. FiLrer
1. Introduction ................................................................ 287
2. Singular bad instances ...................................................... 288
3. Bad instances for any k == 0 (mod 10) ..................................... 292
4. The general case ............................................................ 295
References ..................................................................... 296
Block Exchange in Graph Partitioning ................................... 298
w.
W. Hager, S. C. Park, T. A. Davis
1. Introduction ................................................................ 299
2. Quadratic programming formulation ........................................ 299
3. Numerical illustrations .................................................. ; ... 303
References ..................................................................... 307
On the Efficient Approximability of "HARD"
Problems: A Survey ........................................................ 308
H. B. Hunt III, M. V. Marathe, R. E. Steams
1. Introduction ................................................................ 309