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CuuDuongThanCong.com Handbook of Approximation Algorithms and Metaheuristics CuuDuongThanCong.com CHAPMAN & HALL/CRC COMPUTER and INFORMATION SCIENCE SERIES Series Editor: Sartaj Sahni PUBLISHED TITLES ADVERSARIAL REASONING: COMPUTATIONAL APPROACHES TO READING THE OPPONENT’S MIND Alexander Kott and William M McEneaney DISTRIBUTED SENSOR NETWORKS S Sitharama Iyengar and Richard R Brooks DISTRIBUTED SYSTEMS: AN ALGORITHMIC APPROACH Sukumar Ghosh FUNDAMENTALS OF NATURAL COMPUTING: BASIC CONCEPTS, ALGORITHMS, AND APPLICATIONS Leandro Nunes de Castro HANDBOOK OF ALGORITHMS FOR WIRELESS NETWORKING AND MOBILE COMPUTING Azzedine Boukerche HANDBOOK OF APPROXIMATION ALGORITHMS AND METAHEURISTICS Teofilo F Gonzalez HANDBOOK OF BIOINSPIRED ALGORITHMS AND APPLICATIONS Stephan Olariu and Albert Y Zomaya HANDBOOK OF COMPUTATIONAL MOLECULAR BIOLOGY Srinivas Aluru HANDBOOK OF DATA STRUCTURES AND APPLICATIONS Dinesh P Mehta and Sartaj Sahni HANDBOOK OF SCHEDULING: ALGORITHMS, MODELS, AND PERFORMANCE ANALYSIS Joseph Y.-T Leung THE PRACTICAL HANDBOOK OF INTERNET COMPUTING Munindar P Singh SCALABLE AND SECURE INTERNET SERVICES AND ARCHITECTURE Cheng-Zhong Xu SPECULATIVE EXECUTION IN HIGH PERFORMANCE COMPUTER ARCHITECTURES David Kaeli and Pen-Chung Yew CuuDuongThanCong.com +DQGERRNRI$SSUR[LPDWLRQ $OJRULWKPVDQG0HWDKHXULVWLFV Edited by 7HRÀOR)*RQ]DOH] 8QLYHUVLW\RI&DOLIRUQLD 6DQWD%DUEDUD86$ CuuDuongThanCong.com Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2007 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed in the United States of America on acid-free paper 10 International Standard Book Number-10: 1-58488-550-5 (Hardcover) International Standard Book Number-13: 978-1-58488-550-4 (Hardcover) This book contains information obtained from authentic and highly regarded sources Reprinted material is quoted with permission, and sources are indicated A wide variety of references are listed Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Library of Congress Cataloging-in-Publication Data Handbook of approximation algorithms and metaheurististics / edited by Teofilo F Gonzalez p cm (Chapman & Hall/CRC computer & information science ; 10) Includes bibliographical references and index ISBN-13: 978-1-58488-550-4 ISBN-10: 1-58488-550-5 Computer algorithms Mathematical optimization I Gonzalez, Teofilo F II Title III Series QA76.9.A43H36 2007 005.1 dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com CuuDuongThanCong.com 2007002478 C5505 C5505˙fm April 7, 2007 13:21 DEDICATED To my wife Dorothy, and my children Jeanmarie, Alexis, Julia, Teofilo, and Paolo CuuDuongThanCong.com v C5505 C5505˙fm April 7, 2007 CuuDuongThanCong.com 13:21 vi C5505 C5505˙fm April 7, 2007 13:21 Preface Forty years ago (1966), Ronald L Graham formally introduced approximation algorithms The idea was to generate near-optimal solutions to optimization problems that could not be solved efficiently by the computational techniques available at that time With the advent of the theory of NP-completeness in the early 1970s, the area became more prominent as the need to generate near optimal solutions for NP-hard optimization problems became the most important avenue for dealing with computational intractability As it was established in the 1970s, for some problems one can generate near optimal solutions quickly, while for other problems generating provably good suboptimal solutions is as difficult as generating optimal ones Other approaches based on probabilistic analysis and randomized algorithms became popular in the 1980s The introduction of new techniques to solve linear programming problems started a new wave for developing approximation algorithms that matured and saw tremendous growth in the 1990s To deal, in a practical sense, with the inapproximable problems there were a few techniques introduced in the 1980s and 1990s These methodologies have been referred to as metaheuristics There has been a tremendous amount of research in metaheuristics during the past two decades During the last 15 or so years approximation algorithms have attracted considerably more attention This was a result of a stronger inapproximability methodology that could be applied to a wider range of problems and the development of new approximation algorithms for problems in traditional and emerging application areas As we have witnessed, there has been tremendous growth in field of approximation algorithms and metaheuristics The basic methodologies are presented in Parts I–III Specifically, Part I covers the basic methodologies to design and analyze efficient approximation algorithms for a large class of problems, and to establish inapproximability results for another class of problems Part II discusses local search, neural networks and metaheuristics In Part III multiobjective problems, sensitivity analysis and stability are discussed Parts IV–VI discuss the application of the methodologies to classical problems in combinatorial optimization, computational geometry and graphs problems, as well as for large-scale and emerging applications The approximation algorithms discussed in the handbook have primary applications in computer science, operations research, computer engineering, applied mathematics, bioinformatics, as well as in engineering, geography, economics, and other research areas with a quantitative analysis component Chapters and present an overview of the field and the handbook These chapters also cover basic definitions and notation, as well as an introduction to the basic methodologies and inapproximability Chapters 1–8 discuss methodologies to develop approximation algorithms for a large class of problems These methodologies include restriction (of the solution space), greedy methods, relaxation (LP and SDP) and rounding (deterministic and randomized), and primal-dual methods For a minimization problem P these methodologies provide for every problem instance I a solution with objective function value that is at most (1 + ) · f ∗ (I ), where is a positive constant (or a function that depends on the instance size) and f ∗ (I ) is the optimal solution value for instance I These algorithms take polynomial time with respect to the size of the instance I being solved These techniques also apply to maximization vii CuuDuongThanCong.com C5505 C5505˙fm April 7, 2007 viii 13:21 Preface problems, but the guarantees are different Given as input a value for and any instance I for a given problem P , an approximation scheme finds a solution with objective function value at most (1 + )· f ∗ (I ) Chapter discusses techniques that have been used to design approximation schemes These approximation schemes take polynomial time with respect to the size of the instance I (PTAS) Chapter 10 discusses different methodologies for designing fully polynomial approximation schemes (FPTAS) These schemes take polynomial time with respect to the size of the instance I and 1/ Chapters 11–13 discuss asymptotic and randomized approximation schemes, as well as distributed and randomized approximation algorithms Empirical analysis is covered in Chapter 14 as well as in chapters in Parts IV–VI Chapters 15–17 discuss performance measures, reductions that preserve approximability, and inapproximability results Part II discusses deterministic and stochastic local search as well as very large neighborhood search Chapters 21 and 22 present reactive search and neural networks Tabu search, evolutionary computation, simulated annealing, ant colony optimization and memetic algorithms are covered in Chapters 23–27 In Part III, I discuss multiobjective optimization problems, sensitivity analysis and stability of approximations Part IV covers traditional applications These applications include bin packing and extensions, packing problems, facility location and dispersion, traveling salesperson and generalizations, Steiner trees, scheduling, planning, generalized assignment, and satisfiability Computational geometry and graph applications are discussed in Part V The problems discussed in this part include triangulations, connectivity problems in geometric graphs and networks, dilation and detours, pair decompositions, partitioning (points, grids, graphs and hypergraphs), maximum planar subgraphs, edge disjoint paths and unsplittable flow, connectivity problems, communication spanning trees, most vital edges, and metaheuristics for coloring and maximum disjoint paths Large-scale and emerging applications (Part VI) include chapters on wireless ad hoc networks, sensor networks, topology inference, multicast congestion, QoS multimedia routing, peer-to-peer networks, data broadcasting, bioinformatics, CAD and VLSI applications, game theoretic approximation, approximating data streams, digital reputation and color quantization Readers who are not familiar with approximation algorithms and metaheuristics should begin with Chapters 1–6, 9–10, 18–21, and 23–27 Experienced researchers will also find useful material in these basic chapters We have collected in this volume a large amount of this material with the goal of making it as complete as possible I apologize in advance for omissions and would like to invite all of you to suggest to me chapters (for future editions of this handbook) to keep up with future developments in the area I am confident that research in the field of approximations algorithms and metaheuristics will continue to flourish for a few more decades Teofilo F Gonzalez Santa Barbara, California CuuDuongThanCong.com C5505 C5505˙fm April 7, 2007 13:21 About the Cover The four objects in the bottom part of the cover represent scheduling, bin packing, traveling salesperson, and Steiner tree problems A large number of approximation algorithms and metaheuristics have been designed for these four fundamental problems and their generalizations The seven objects in the middle portion of the cover represent the basic methodologies Of these seven, the object in the top center represents a problem by its solution space The object to its left represents its solution via restriction and the one to its right represents relaxation techniques The objects in the row below represent local search and metaheuristics, problem transformation, rounding, and primal-dual methods The points in the top portion of the cover represent solutions to a problem and their height represents their objective function value For a minimization problem, the possible solutions generated by an approximation scheme are the ones inside the bottommost rectangle The ones inside the next rectangle represent the one generated by a constant ratio approximation algorithm The top rectangle represents the possible solution generated by a polynomial time algorithm for inapproximable problems (under some complexity theoretic hypothesis) ix CuuDuongThanCong.com C5505 C5505˙ind March 20, 2007 21:11 IN-11 Index mathematical models, 69-2 loss/delay, 69-3 tree, 69-2 multicast traffic, 69-1 performance, 69-1 resources, 69-1 topology inference, 69-2 A-approach, 69-4 accuracy, 69-7 binary Hamming distance approach, 69-4, 69-6 evaluation, 69-9 general trees, 69-12 Hamming distance approach, 69-4, 69-5 neural nets feedforward and recurrent, 22-1 neural networks, 1-9 artificial, 22-1–22-12 Newton’s method, 30-5 next fit, 32-2 next fit decreasing, 35-2 niching, 24-7 no free lunch theorem, 27-2 no-fit polygon, 36-11 node disjoint paths, 64-2 NP-completeness, 1-4, 1-11–1-12 NPO, 15-2 NPO-complete problems, 15-6, 15-7 NPO-completeness, 15-7, 15-10, 16-10 NPO-DPB, 16-13 NPO-PBNPO-PB, 15-13 numerical computing, 60-3 O object location problem, 72-11 objective function, 14-7, 18-2, 30-2 oblivious shifting, 9-9 OCT, see optimum communication spanning tree offline bin packing, 32-2, 35-6 packing, 35-4 online bin packing, 32-2, 35-6 packing, 35-3 operation count, 14-3 optimality max, 28-2 hierarchical, 28-2 Pareto, 28-2 optimality notions, 29-2–29-3 Pareto, 29-2 scalarized, 29-2 global optimum, 29-3 optimization constrained, 24-8 optimum communication spanning trees, 59-1 p-source, 59-2, 59-15 cost function, 30-2 product-requirement communication spanning trees, 59-2, 59-8 CuuDuongThanCong.com sum-requirement communication spanning trees, 59-2, 59-10 optional subtask, 44-1 oracle, 30-6 orienteering problem, 40-5 outer verifier, 17-12 overlay network, 72-1 decentralized, 72-8 supervised, 72-5 P p-covering, 16-4 packing, 2-13, 37-1, 78-1 congruent, 78-1 feasible, 37-5 job interval, 37-2 lattice, 78-2 rectangles, 35-11 sphere, 78-1 Annuli minimization, 78-7 containing region, 78-5 lattice, 78-2 pack-and-shake, 78-2 rotational lattice, 78-6 shaking, 78-8 thickest point, 78-4 translational lattice, 78-4 trimming, 78-9 squares, 35-11 weighted set, 37-4 packing integer programs, 57-9 column-restricted, 57-9, 57-11 for unsplittable flow, 57-9 pairwise local sequence alignment, 76-3 parallel algorithms, 32-15, 62-1, 62-6, 62-10, 62-13 evolutionary algorithms, 24-10 genetic algorithm, 24-4 parallelism adaptive, 41-2 maximum, 41-9 parameter sensitivity analysis, 14-11 space decomposition, 30-2, 30-5 tuning, 21-2 parameterized intractability, 74-5 parametric analysis, 30-1 ancestral reconstruction, 30-8 linear programming, 30-8 matroid optimization, 30-9 maximum flow, 30-10 Newton’s method, 30-5 optimum subgraph problems on planar graphs, 30-12 problems construction, 30-7 parametric search, see bisection search, gradient descent, Megiddo’s method search, 30-4 simplex method, 30-8 stable marriage, 30-7 C5505 C5505˙ind March 20, 2007 IN-12 21:11 Handbook of Approximation Algorithms and Metaheuristics Pareto curve, 28-2 approximate, 28-10 set, 28-2 approximate, 28-10 Pareto-optimal solution, 14-14 partial constraint satisfaction, 76-7 long alignments high normalized scores, 76-11 length threshold, 76-7 partial cover, 5-2 partial enumeration, 9-2 partition, 3-6, 54-1, 61-1 adaptive, 42-7 balance, see balance constraints guillotine cutting stock, 54-14 multidimensional, 3-6 multidimensional space, 54-14 optimal, 3-6, 54-2, 54-8 suboptimal, 3-6, 54-2 rectangle, 3-6 edge length, 3-6 rectangular, 54-1 edge length, 3-6 minimum edge length, 54-1 multidimensional, 3-6 refinement, 61-14 tolerance, see balance constraints partition(s), 61-4, 61-5 partitioning, see hypergraph partitioning flat, 61-5 multilevel hypergraph, 79-6 multiway, 79-7 VLSN search, 20-3 partitioning graphs, 2-15 partitionment k-way, 61-4 partitions, 55-1 guillotine, 42-8 rectangular, 42-8 path coloring, 57-9 path relinking, 48-9, 48-10 PCP, 17-6 PCP theorem, 17-1, 17-5 peer-to-peer systems, 2-16, 72-1 penalty traveling salesperson problem, 40-2 penalty weight, 48-5 perceptrons, 22-3 performance evaluation, 1-12–1-15 persistent dynamic sets, 21-6 personalized medicine, 74-2 perturbation, 51-4 phenotype space, 24-2 pheromone, 26-2, 64-4 evaporation, 26-5 trail, 26-2, 26-3 update, 26-5 physical design, 79-1 piecewise linear algorithm, 30-6 CuuDuongThanCong.com linear function, 30-2 polynomials, 84-3, 84-8 pivoting rule, 19-4 best-improvement, 19-4 first-improvement, 19-4 placement, 79-1 analytical, 79-5, 79-8, 79-11, 79-12 assignment, 79-5 constructive, 79-4 continuous, 38-3 discrete, 38-3 fixed terminals, 79-5 force-directed, 79-5 fractional cut, 79-7 genetic algorithms, 79-4 global, 79-2 grid warping, 79-8 integer nonlinear-programming formulation, 79-3 iterative, 79-4 feasible, 79-4 infeasible, 79-5 legalization, 79-5 Monte Carlo, 79-4 multiscale (multilevel), 79-10 net-weighting, 79-13 network flow, 79-5 nonlinear-programming formulation, 79-12 partitioning-based, 79-8 path-based, 79-13 performance-driven, 79-13 Poisson-based, 79-6, 79-12 relaxation, 79-11 routability, 79-14 simulated annealing, 79-4 synthetic benchmarks, 79-14 timing-driven, 79-13 top-down partitioning-based, 79-6 unconstrained quadratic, 79-5 wirelength estimate log-sum-exp, 79-11 wirelength estimation bounding-box, 79-3 planar subgraph maximum weight, 2-15 planning, 47-1 actions, 47-2 automated, 2-14 complexity, 47-2 domain model, 47-1 heuristics, 47-4 lagrange multipliers, 47-10, 47-13 linear programming, 47-7 local search, 47-4, 47-10 local search plan graph, 47-11 makespan, 47-2 orienteering problem, 47-9 plan graph, 47-5 plan space search, 47-4 relaxation, 47-4 C5505 C5505˙ind March 20, 2007 21:11 IN-13 Index satisfiability, 47-12 state space search, 47-4 STRIPS, 47-2 planted motif search, 2-16 PLS, 20-10 complexity class, 18-3 Poly-APX, 15-4, 16-12 Poly-APX-canonically hard, 16-12 Poly-DAPX, 16-11, 16-12 Poly-DAPX-complete, 16-11, 16-12 polymatroid matching, 56-2 polynomial time approximation scheme, see PTAS polynomial time local search problems, 18-3 population-based metaheuristics, 27-2 portals, 42-10 positive semidefinite matrix, 8-6 power assignment, 67-1 precisely scheduled task, 44-2 prediction presidential election, 74-4 prefix technique, 72-4 preprocessing, 63-3 preselection, 24-7 price of anarchy, 81-1, 81-3, 83-1 price of stability, 81-7 primal-dual, 2-11–2-12, 4-6–4-8, 13-14–13-17, 37-4–37-14, 40-2–40-12, 71-10–71-14 primal-dual methods, 4-6, 39-2, 71-11 primer selection, 2-16, 75-1 degenerate, 75-1 prior analysis, 30-4 Prisoner’s Dilemma, 85-4 prize collecting Steiner tree problem, 40-2 prize-collecting TSP, 40-9, 47-9 probabilistic algorithm, 49-8 domination, 14-5, 14-10 iterative improvement, 19-5 oracle machines, 17-6 proof systems, 17-6 rounding, 11-13 probabilistically checkable proofs, see PCP PROCT, see optimum product-requirement communication spanning tree promise problems, 17-4 proximity problems, 2-14 proxy, 41-4 PRR, 72-11 PRR scheme, 72-11 neighbor sequence, 72-13 neighbor table, 72-12 pointer, 72-12 prefix routing, 72-12 random label, 72-11 Route, Join, and Leave, 72-13 virtual identifier, 72-11 pseudo-approximation, 51-11 pseudorandom proportional rule, 26-9 CuuDuongThanCong.com PTAS, 1-15, 9-1–9-19, 15-3, 17-3, 42-7, 42-10, 46-5, 51-3, 51-11, 59-6, 59-9, 59-13, 59-15, 70-7, 75-15, 75-21, 81-7, 83-17 knapsack, 9-4 scheduling identical machines, 9-7 PTAS, 15-3, 15-8, 15-13, 16-12–16-14 PTAS-complete, 15-14 PTAS-reducibility, 15-15, 16-12 pure exchange economy, 82-2 push neighborhood, 18-5 Q QoS, 71-1 multicast, 71-1 multicast tree, 71-2 quadratic optimization, 8-1 programming, 8-1 quadratic optimization, 8-4 qualified run-time distribution (QRTD), 14-8 Quality of Service, 10-8, see QoS quasi-greedy algorithm, 39-10 quota traveling salesperson problem, 40-5 R radial problem, 16-9 radiocoloring, 12-7 order, 12-7 span, 12-7 radiosurgery, 78-2 random hyperplane rounding, 8-2 moves, 24-5 randomized algorithms, 62-1, 62-6 empitical evaluation, 14-1–14-15 approximation techniques, 12-1–12-11 dissection, 9-17 grouping, 9-9 iterative improvement, 19-5 local search, 25-1 rounding, 4-9, 6-6–6-10, 7-2–7-5, 12-2, 49-8, 70-7–70-10, 80-8 rapid mixing, 12-8 rational player, 85-4 ray shooting, 30-4 reactive search, 1-9, 21-1–21-14 escape mechanism, 21-4 real-time BPTT, 22-10 recombination, 24-2 rectangle packing, 36-1, 36-2, 36-6, 36-9 area minimization, 36-3 bin packing, 36-3 cutting stock, 36-3 knapsack, 36-3 pallet loading, 36-3 strip packing, 36-3 rectangle stabbing, 7-9 rectangle stretching, 35-10 rectangular partitions, 2-14, 42-8 C5505 C5505˙ind March 20, 2007 IN-14 21:11 Handbook of Approximation Algorithms and Metaheuristics rectilinear, 79-14 distance facility location (RDFL), 79-5 partitioning, 7-10 recursive greedy algorithm, 5-7 recursive largest first, 63-4 red-black trees, 21-6 limited node copying, 21-7 path copying, 21-7 reduced cost, 48-7 reducibility, 15-1 AP, 15-8, 15-9 DFT, 16-13 DPTAS, 16-11, 16-12 D, 16-10, 16-11 E, 15-8, 16-12 FT, 15-13, 16-13 GAP, 16-10 G, 16-10, 16-11 L, 15-5 PTAS, 15-15, 16-12 S-D, 16-12 S, 15-6 basic, 15-4 Karp, 15-1 linear, 15-5 strict, 15-6, 16-10 Turing, 15-13 reductions preserve approximability, 15-1–15-15 refinement helpful set, 60-10 relative cost, 48-7 relative neighborhood graph, 68-3 relaxation, 3-1, 30-12, 47-4, 79-10 LP, 2-9 relaxation-based local search, 79-5 relaxed triangle inequality, 31-4 replication, 79-7 reproducibility, 14-2 reproduction operators, 24-3 reputation context, 85-9 second-order, 85-7 reputation in virtual communities, 2-17 resource allocation, 45-1 resource augmentation, 35-7 restricted shortest path, 10-8 restriction methods, 2-1–2-5, 3-1–3-10, 30-12 Rietz’s identity, 8-3 ring heuristic, 50-1, 50-6 ring networks, 70-1 robust tag SNPs, 2-17 robustness, 30-4 root-finding, 30-4 rotations, 35-5 rounding, 2-9–2-10, 6-2–6-6, 7-1, 7-5, 7-10, 9-6, 9-11, 10-1–10-5, 10-9–10-12, 35-4, 45-5–45-14, 46-7, 57-3–57-12, 58-13, 70-1–70-7 deterministic, 7-5 down, 10-1 random, 10-1 CuuDuongThanCong.com randomized, 7-2, 10-2 up, 10-1 routability, 79-14 routing, 79-1 load, 59-3 product-requirement, 59-9 sum-requirement, 59-11 nets layout area, 3-7 multiterminal, 3-7, 3-9 two-terminal, 3-7, 3-9 run-time, 14-2 environment, 14-2 run-time distribution (RTD), 14-3, 14-15 bivariate, 14-8 S S-D-reducibility, 16-12 S-reducibility, 15-6 sample size, 14-5 SAT, 2-14, 14-2, 14-7, 16-10, 59-12 max, 16-3, 49-1 min, 16-3, 49-1 satisfiability problem, see SAT saturation degree, 63-4 scalable, 79-2, 79-6, 79-10 scaling, 10-1 scaling analysis, 14-11 scaling and rounding, 59-10 scatter search, 1-9, 24-8, 27-1 VLSN search, 20-2 schedules cooling, 25-3 dynamic, 25-4 geometric, 25-4 static, 25-4 scheduling, 28-9, 28-11, 81-1, 81-7 anomalies, 1-3 average expected delay (AED), 73-4 biobjective, 28-5 broadcast nonuniform allocation, 73-4 uniform allocation, 73-4 data broadcasts, 2-16, 73-2 identical machines, 4-5, 46-6 assign any task, 2-3 critical path, 2-5 deadline, 4-5 FPTAS, 2-8 independent tasks, 2-5 list, 2-5 list scheduling, 1-2 LPT, 2-5 makespan, 2-2, 9-7 max profit, 4-5 multifit, 2-8 no-additional-delay, 2-3 optimal makespan, 2-2 precedence constraints, 1-2, 2-3 PTAS, 1-4, 2-8 release time, 4-5 C5505 C5505˙ind March 20, 2007 21:11 Index imprecise computation model, 2-14 jobs, 83-5 list, 45-2 LPT, 81-6 LPT-SPT, 81-6 makespan, 81-6, 81-7 malleable task, 2-14, see malleable task scheduling malleable tasks -approximation, 45-3 AFPTAS, 45-5 allotment, 45-2 bounded width graph, 45-8 identical machines, 45-2 series parallel graph, 45-8 trees, 45-8 moldable task, see malleable task scheduling multiprocessor, 16-8, 18-1 makespan, 18-1 periodic, 73-4 PTAS, 81-7 SPT, 81-6, 81-10 unrelated machines, 6-3 LP rounding, 6-3 makespan, 6-3 vehicle, 2-14, see vehicle scheduling school-bus driver problem, 40-10 scientific computing, 60-3 SDP relaxation, 8-2 search cost distribution (SCD), 14-4, 14-15 search model CWAC, 29-2 SAC, 29-2 search path storage, 21-5 search space pruning, 47-4 search tree, 41-1, 41-6 second greedy algorithm, 77-8 selection, 24-2 self-healing, 41-4 selfish agents, 81-2 selfish machines, 81-2 semi-online bin packing, 32-6 semidefinite programming (SDP), 2-1, 8-1–8-15, 49-8 sensitivity analysis, 1-10, 30-1–30-13 prior analysis, 62-1 shortest paths, 30-9 sensitivity of heuristics, 30-13 knapsack, 30-13 scheduling, 30-13 subset sum, 30-13 sensitivity of minimum spanning tree, 30-2 sensitivity of traveling salesman tour, 30-2 separation, 10-1, 10-6–10-7, 10-11 separation lemma, 82-8 separator, 59-4 separator/shredder, 58-10 set packing, 16-12, 37-4, 37-8 set-cover, 1-7, 4-1, 5-2, 7-2, 12-2, 16-5, 16-12, 58-18, 75-4, 83-5 k-set cover, 16-5 unweighted, 4-9 CuuDuongThanCong.com IN-15 set-family laminar, 58-7, 58-13, 58-16 set-function crossing supermodular, 58-6, 58-13 setpair-function, 58-5 skew-supermodular, 58-6, 58-15 weakly supermodular, 58-6 Shared, 41-6, 41-7 sharpened triangle inequality, 31-4 shelf algorithms, 35-3 shifting, 9-6 plane, 9-18 shortest paths, 10-1, 28-4, 28-6 k most vital edges, 62-1 forest, 59-12, 59-16 tree, 59-3, 59-11, 59-14 multiconstrained, 10-7–10-14 restricted, 10-8 routing, 11-13 shortest superstring, 4-2 simple composition technique, 83-8 simple problem, 16-8 simulated annealing, 1-9, 19-5, 24-10, 25-1–25-10 placement, 79-4 single nucleotide polymorphism, 77-1 auxiliary tag SNP, 77-3 coding SNP, 77-2 missing data, 77-3 nonsynonymous SNP, 77-2 robust tag SNP, 77-4 synonymous SNP, 77-2 tag SNP, 77-2 skeleton, 59-15 reduced, 59-15 slack, 79-14 SNP, see single nucleotide polymorphism social welfare, 81-9 software for hypergraph partitioning, 61-15 hMETIS, 61-15 MLPart, 61-15 solution approximate, 28-6, 28-8 basic, 58-6, 58-13, 58-16 best-so-far, 26-6 decomposition, 59-3 efficient, 28-2 iteration best, 26-6 quality, 14-7 relative, 14-8, 14-11 quality distribution (SQD), 14-9, 14-15 quality over time (SQT) curve, 14-8, 14-9 quality trace, 14-8 supported, 28-5, 29-3 Solution interface, 41-7 space decision, 28-2 objective, 28-2 space complexity, 1-10 spanner, 53-7, 68-1 length spanner, 68-2 power spanner, 68-2 C5505 C5505˙ind March 20, 2007 IN-16 21:11 Handbook of Approximation Algorithms and Metaheuristics spanning trees, 3-2, 28-4, 59-1, 71-3 k most vital edges, 62-1 efficient communication, 2-15 maximum leaves, 18-7 minimum-degree, 4-5 sparsified Yao graph, 68-6 speedup heterogeneous, 41-8 homogeneous, 41-8 sphere packing medical applications, 2-17 split tree, 53-3 splittable flow, 57-12 spoke triangulation, 50-8 square packing, 9-18 square root rule, 73-2 SROCT, see optimum sum-requirement communication spanning tree stabbing, 37-1 job interval, 37-2 rectangle, 37-4 stability, 1-10, 30-3, 31-4 stability of -approximate solutions, 30-5 stability of approximation, 31-1–31-13 stability of heuristics, 30-5 stability radius, 30-4 stability region, 30-4 stagnation detection, 14-12 standard sigmoid function, 22-3 star, 59-4 k-star, 59-6 configuration, 59-7 general, 59-4 state space search progression, 47-4 regression, 47-4 static schedules, 25-4 stationary distribution, 25-6 statistical tests, 14-5, 14-7, 14-13, 14-15 Steiner Euclidean k-connectivity, 51-2 Steiner network generalized LP rounding, 6-5 Steiner ratio, 42-2 Banach spaces, 42-4 Euclidean space, 42-4 Steiner trees, 1-3, 2-14, 3-2, 4-3, 42-1, 43-1, 43-11, 51-3, 51-5, 56-6, 65-3, 70-2, 71-1, 79-14 2-restricted, 42-3 3-restricted, 42-3, 42-5 k-restricted, 42-3, 42-5, 71-4 β-convex, 71-4 -shallow k-DST, 5-5 k-DST, 5-5 bottleneck edges, 43-8, 43-9 directed, 5-4 directed Steiner tree (DST), 5-5 Euclidean, 42-2 full, 42-3, 43-3 CuuDuongThanCong.com geometric, 42-7 group Steiner tree (GS), 5-12 heuristics, 65-3 k-restricted, 56-6 multicast, 70-1 networks, 42-2 node-weighted, 4-3 preprocessing, 42-7 rectilinear, 42-4 triple contraction algorithm, 43-3, 43-5 stigmergy, 26-2 stochastic local search constructive algorithms, 19-3 definition, 19-1 evaluation function, 19-2 step function, 19-2 stochastic problem, 14-14 storage allocation, 32-1 stretch factor, 52-1 strict reduction, 16-10 strip packing, 35-3, 45-2 three-dimensional, 35-1 two-dimensional, 35-1 subgradient, 30-6 subgradient method, 48-6 subtree cover, 46-1 minmax, 46-2 MSCP-TREE, 46-2 rooted tree, 46-2 rooted, 46-2 survivability, 64-2 survivable network design, 51-2 swap neighborhood, 18-5 swap operation, 18-10 swarm intelligence, 26-1 Sybil attack, 85-10 symmetric Yao graph, 68-6 synthetic benchmarks, 79-14 systems autonomous, 83-1 heterogeneous, 83-1 T Tabu cycle method, 23-10 Tabu search, 1-9, 19-6, 23-1, 24-10 adaptive memory, 23-1 applications, 23-2 constructive neighborhoods, 23-4 destructive neigborhoods, 23-4 diversification strategy, 23-5 features, 23-2 frequency-based memory, 23-4 global optimum, 23-3 intensification strategy, 23-5 local optimum, 23-3 modified neighborhood, 23-3 move, 23-3 neighborhood, 23-3 path relinking, 23-6 penalty function, 23-6 prohibition period, 21-4 C5505 C5505˙ind March 20, 2007 21:11 IN-17 Index recency-based memory, 23-4 residence measure, 23-5 responsive exploration, 23-1 stratigic oscillation, 23-6 tabu-active attributes, 23-5 transistion measure, 23-5 VLSN search, 20-2 Tabu tenure, 19-6 targeted therapies, 74-2 Task, 41-2, 41-4, 41-7 atomic, 41-6 caching, 41-5 executed by task server, 41-6 prefetching, 41-5 task server, 41-4 taxonomy cancer, 74-11 television programming, 32-1 terminal propagation, 79-7 termination time distribution (TTD), 14-9 tetrahedralization 3D, 50-1 Tetris-like packing, 35-10 threshold circuits, 22-4 threshold functions, 55-11 S-threshold functions, 55-13 binary, 22-3 Chow parameters, 55-12 enumeration, 55-11 tight inequality, equation, 58-6 set, 58-7 time complexity, 1-10 time-cost tradeoff, 45-11 budget, 45-11 timing, 79-13 tolerance, 30-4, see balance constraints lower, 30-4 upper, 30-4 topology control, 2-16, 67-1, 68-2 topology discovery, 2-16 total error, 44-1 traditional applications, 2-13–2-14 tragedy of the commons, 85-4 transformation-restriction, 3-9 transitive closure, 5-5 transportation, 32-1 traveling repairman problem, 40-10 traveling salesperson, 3-3, 14-2, 17-3, 28-3, 28-12, 31-4, 31-8, 40-1, 41-1, 51-3 p salespeople, 46-1 S -algorithm, 31-8 T -algorithm, 31-7 approximability lower bounds, 31-11 asymmetric, 17-3, 31-12 Christofides algorithm, 31-5 cycle cover algorithm, 31-10 double spanning tree algorithm, 31-5 non-metric, 17-3 PMCA algorithm, 31-9 CuuDuongThanCong.com problem, 14-7 tour, 51-12 TSPLIB, 41-8 trees, 45-8 k-restricted Steiner, 56-6 Steiner, 56-6 triangle inequality, 18-10 parameterized, 31-4 relaxed, 31-4 sharpened, 31-4, 31-10 triangular cactus, 56-4 structure, 56-4 triangulation 2D, 50-1 constrained Delaunay, 50-2 Delaunay, 50-2 greedy, 50-2 max weight, 50-2, 50-8 weight, 50-2, 50-4 triangulations, 2-14 Delaunay, 42-5 trust institution-based, 85-2 kinds of, 85-1 transaction-based, 85-2 truthful algorithms, 81-9 TS, 23-1 TSP, 1-3, 2-14, 3-3, 16-3, 16-7, 16-8, 26-6, see traveling salesperson, 31-12, 40-1, 41-1 metric, 3-3, 16-8 U uniform orientation metric, 43-1, 43-11 UNION-FIND algorithm, 44-4 unit-disk graph, 53-11, 68-2 unit-disk graph model, 65-1 unrolling a recurrent net, 22-8 unsplittable flow, 64-2 balance condition, 57-7 combinatorial algorithms, 57-10 definition, 57-1 hardness of approximation, 57-5, 57-10 high-capacity case, 57-9 integrality gaps, 57-9 LP-rounding algorithms, 57-3, 57-7–57-10 maximizing demand, 57-8–57-10 minimizing congestion, 57-8 no-bottleneck, see balance condition57-7 randomized rounding for, see LP-rounding algorithms routing in rounds, 57-9 single-source, 57-11 weighted, 57-7 user as a learning component, 21-1 utilitarian objective function, 83-2 utility functions, 82-6 CES, 82-6 Cobb–Douglas, 82-9 concave homogeneous, 82-6 C5505 C5505˙ind March 20, 2007 IN-18 21:11 Handbook of Approximation Algorithms and Metaheuristics utility functions (continued) Leontief, 82-13 linear, 82-4 log-linear, 82-11 Uzawa method, 79-12 V valid inequality, 48-11 valuation function, 83-3 variable neighbourhood descent, 19-4 variable neighbourhood search, 19-4 VC dimensions, 22-6 VC-dimension bounds of neural nets, 22-6 VCG mechanism, 81-10 vector packing, 35-9 vector scheduling, 9-10 vehicle scheduling, 46-1 1-VSP-PATH, 46-1 1-VSP-TREE, 46-1 connected schedule, 46-5 depth-first schedule, 46-3 eager schedule, 46-5 gapless schedule, 46-7 makespan, 46-7 VSP-PATH, 46-1 VSP-TREE, 46-1 zone schedule, 46-7 vertex cover, 4-6, 7-5, 13-14, 16-1, 16-3, 16-4, 16-12, 16-13, 49-2, 74-3, 83-10 α-vector approach, 2-10 greedy method, 2-8 local ratio approach, 2-11 LP, 6-1 LP relaxation, 6-1 primal-dual approach, 2-11 rounding, 6-2 weighted, 83-10 LP-relaxation, 2-8 vertex matching, 79-10 very large-scale neighborhood search, 20-2 violated subset, 37-5 virtual community definition, 85-1 virtual identities multiple, 85-2 virtual infrastructure clustering techniques, 2-15 virtual space, 72-2 CuuDuongThanCong.com VLSI, 79-1 global routing, 2-17 physical design, 43-1 placement, 2-17 VLSN, 20-2 VLSN search, 1-9 applications, 20-2 capacitated spanning trees, 20-3 cyclic exchange, 20-3 cyclic exchange neighborhood, 20-3 domination analysis, 20-7 extended neighborhood, 20-7 generalized assignment, 20-3 genetic algorithms, 20-2 graph partioning, 20-3 k-constrained multiple knapsack, 20-3 multiexchange neighborhood, 20-3 path exchange neighborhood, 20-3 scatter search, 20-2 tabu search, 20-2 vehicle routing, 20-3 VSP, see vehicle scheduling W wavelets, 84-3, 84-10 restricted optimizations, non– error, 84-13 unrestricted optimizations, error, 84-11 unrestricted optimizations, non– error, 84-12 weak duality, 4-6 weak duality lemma, 37-3 weight vector λ, 29-3 weighted sum, 29-3 well-separated pair decomposition, 2-14, 53-3 WHS, 37-11 wireless ad hoc networks, 68-1 wireless sensor network, 65-1 face routing, 65-10 geographic routing, 65-10 worst fit, 32-3 worst solution, 16-2, 16-3, 16-12 worst value, 16-2 wsat, 16-10, 16-12 WSPD-spanner, 52-4 X X-architecture, 43-1 Y Y-architecture, 43-1 Yao graph, 68-6 C5505 C5505˙ind April 11, 2007 CuuDuongThanCong.com 13:20 19 C5505 C5505˙ind April 11, 2007 CuuDuongThanCong.com 13:20 20 C5505 C5505˙ind April 11, 2007 CuuDuongThanCong.com 13:20 21 C5505 C5505˙ind April 11, 2007 CuuDuongThanCong.com 13:20 22 C5505 C5505˙ind April 11, 2007 CuuDuongThanCong.com 13:20 23 C5505 C5505˙ind April 11, 2007 CuuDuongThanCong.com 13:20 24 CuuDuongThanCong.com ... United States of America on acid-free paper 10 International Standard Book Number-10: 1-5 848 8-5 5 0-5 (Hardcover) International Standard Book Number-13: 97 8-1 -5 848 8-5 5 0-4 (Hardcover) This book contains... information science ; 10) Includes bibliographical references and index ISBN-13: 97 8-1 -5 848 8-5 5 0-4 ISBN-10: 1-5 848 8-5 5 0-5 Computer algorithms Mathematical optimization I Gonzalez, Teofilo F II... Computational Geometry and Graph Applications Large-Scale and Emerging Applications 2-8 2-1 2 2-1 3 2-1 4 2-1 5 Introduction In Chapter we presented an overview of approximation
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