Complexity and Approximation

Combinatorial Optimization Problems and Their Approximability Properties

By: Giorgio Ausiello (Editor), Pierluigi Crescenzi (Editor), Giorgio Gambosi (Editor)

Write A Review

Hardcover | 3 December 2002

At a Glance

Hardcover
548 Pages

Dimensions(cm)
24.77 x 20.32 x 3.81

Hardcover

$139.00

or 4 interest-free payments of $34.75 with

Ships in 5 to 7 business days

N COMPUTER applications we are used to live with approximation. Var- I ious notions of approximation appear, in fact, in many circumstances. One notable example is the type of approximation that arises in numer- ical analysis or in computational geometry from the fact that we cannot perform computations with arbitrary precision and we have to truncate the representation of real numbers. In other cases, we use to approximate com- plex mathematical objects by simpler ones: for example, we sometimes represent non-linear functions by means of piecewise linear ones. The need to solve difficult optimization problems is another reason that forces us to deal with approximation. In particular, when a problem is computationally hard (i. e., the only way we know to solve it is by making use of an algorithm that runs in exponential time), it may be practically unfeasible to try to compute the exact solution, because it might require months or years of machine time, even with the help of powerful parallel computers. In such cases, we may decide to restrict ourselves to compute a solution that, though not being an optimal one, nevertheless is close to the optimum and may be determined in polynomial time. We call this type of solution an approximate solution and the corresponding algorithm a polynomial-time approximation algorithm. Most combinatorial optimization problems of great practical relevance are, indeed, computationally intractable in the above sense. In formal terms, they are classified as Np-hard optimization problems.

Shipping

	Standard Shipping	Express Shipping
Metro postcodes:	$9.99	$14.95
Regional postcodes:	$9.99	$14.95
Rural postcodes:	$9.99	$14.95

Orders over $89.00 qualify for free shipping.

How to return your order

At Booktopia, we offer hassle-free returns in accordance with our returns policy. If you wish to return an item, please get in touch with Booktopia Customer Care.

Additional postage charges may be applicable.

Defective items

If there is a problem with any of the items received for your order then the Booktopia Customer Care team is ready to assist you.

For more info please visit our Help Centre.

You Can Find This Book In

Non-Fiction Computing & I.T.Computer Programming & Software Development Algorithms & Data Structures Reference, Information & Interdisciplinary Subjects Encyclopaedias & Reference Works Reference Works Computer Science Mathematical Theory of Computation Systems Analysis & Design Mathematics Optimisation

Game Theory Algebra Business Applications Databases Data Capture & Analysis Calculus & Mathematical Analysis Numerical Analysis

The Complexity of Optimization Problems	p. 1
Analysis of algorithms and complexity of problems	p. 2
Complexity analysis of computer programs	p. 3
Upper and lower bounds on the complexity of problems	p. 8
Complexity classes of decision problems	p. 9
The class NP	p. 12
Reducibility among problems	p. 17
Karp and Turing reducibility	p. 17
NP-complete problems	p. 21
Complexity of optimization problems	p. 22
Optimization problems	p. 22
PO and NPO problems	p. 26
NP-hard optimization problems	p. 29
Optimization problems and evaluation problems	p. 31
Exercises	p. 33
Bibliographical notes	p. 36
Design Techniques for Approximation Algorithms	p. 39
The greedy method	p. 40
Greedy algorithm for the knapsack problem	p. 41
Greedy algorithm for the independent set problem	p. 43
Greedy algorithm for the salesperson problem	p. 47
Sequential algorithms for partitioning problems	p. 50
Scheduling jobs on identical machines	p. 51
Sequential algorithms for bin packing	p. 53
Sequential algorithms for the graph coloring problem	p. 58
Local search	p. 61
Local search algorithms for the cut problem	p. 62
Local search algorithms for the salesperson problem	p. 64
Linear programming based algorithms	p. 65
Rounding the solution of a linear program	p. 66
Primal-dual algorithms	p. 67
Dynamic programming	p. 69
Randomized algorithms	p. 74
Approaches to the approximate solution of problems	p. 76
Performance guarantee: chapters 3 and 4	p. 76
Randomized algorithms: chapter 5	p. 77
Probabilistic analysis: chapter 9	p. 77
Heuristics: chapter 10	p. 78
Final remarks	p. 79
Exercises	p. 79
Bibliographical notes	p. 83
Approximation Classes	p. 87
Approximate solutions with guaranteed performance	p. 88
Absolute approximation	p. 88
Relative approximation	p. 90
Approximability and non-approximability of TSP	p. 94
Limits to approximability: The gap technique	p. 100
Polynomial-time approximation schemes	p. 102
The class PTAS	p. 105
APX versus PTAS	p. 110
Fully polynomial-time approximation schemes	p. 111
The class FPTAS	p. 111
The variable partitioning technique	p. 112
Negative results for the class FPTAS	p. 113
Strong NP-completeness and pseudo-polynomiality	p. 114
Exercises	p. 116
Bibliographical notes	p. 119
Input-Dependent and Asymptotic Approximation	p. 123
Between APX and NPO	p. 124
Approximating the set cover problem	p. 124
Approximating the graph coloring problem	p. 127
Approximating the minimum multi-cut problem	p. 129
Between APX and PTAS	p. 139
Approximating the edge coloring problem	p. 139
Approximating the bin packing problem	p. 143
Exercises	p. 148
Bibliographical notes	p. 150
Approximation through Randomization	p. 153
Randomized algorithms for weighted vertex cover	p. 154
Randomized algorithms for weighted satisfiability	p. 157
A new randomized approximation algorithm	p. 157
A 4/3-approximation randomized algorithm	p. 160
Algorithms based on semidefinite programming	p. 162
Improved algorithms for weighted 2-satisfiability	p. 167
The method of the conditional probabilities	p. 168
Exercises	p. 171
Bibliographical notes	p. 173
NP, PCP and Non-approximability Results	p. 175
Formal complexity theory	p. 175
Turing machines	p. 175
Deterministic Turing machines	p. 178
Nondeterministic Turing machines	p. 180
Time and space complexity	p. 181
NP-completeness and Cook-Levin theorem	p. 184
Oracles	p. 188
Oracle Turing machines	p. 189
The PCP model	p. 190
Membership proofs	p. 190
Probabilistic Turing machines	p. 191
Verifiers and PCP	p. 193
A different view of NP	p. 194
Using PCP to prove non-approximability results	p. 195
The maximum satisfiability problem	p. 196
The maximum clique problem	p. 198
Exercises	p. 200
Bibliographical notes	p. 204
The PCP theorem	p. 207
Transparent long proofs	p. 208
Linear functions	p. 210
Arithmetization	p. 214
The first PCP result	p. 218
Almost transparent short proofs	p. 221
Low-degree polynomials	p. 222
Arithmetization (revisited)	p. 231
The second PCP result	p. 238
The final proof	p. 239
Normal form verifiers	p. 241
The composition lemma	p. 245
Exercises	p. 248
Bibliographical notes	p. 249
Approximation Preserving Reductions	p. 253
The World of NPO Problems	p. 254
AP-reducibility	p. 256
Complete problems	p. 261
NPO-completeness	p. 261
Other NPO-complete problems	p. 265
Completeness in exp-APX	p. 265
APX-completeness	p. 266
Other APX-complete problems	p. 270
Exercises	p. 281
Bibliographical notes	p. 283
Probabilistic analysis of approximation algorithms	p. 287
Introduction	p. 288
Goals of probabilistic analysis	p. 289
Techniques forthe probabilistic analysis of algorithms	p. 291
Conditioning in the analysis of algorithms	p. 291
The first and the second moment methods	p. 293
Convergence of random variables	p. 294
Probabilistic analysis and multiprocessor scheduling	p. 296
Probabilistic analysis and bin packing	p. 298
Probabilistic analysis and maximum clique	p. 302
Probabilistic analysis and graph coloring	p. 311
Probabilistic analysis and Euclidean TSP	p. 312
Exercises	p. 316
Bibliographical notes	p. 318
Heuristic methods	p. 321
Types of heuristics	p. 322
Construction heuristics	p. 325
Local search heuristics	p. 329
Fixed-depth local search heuristics	p. 330
Variable-depth local search heuristics	p. 336
Heuristics based on local search	p. 341
Simulated annealing	p. 341
Genetic algorithms	p. 344
Tabu search	p. 347
Exercises	p. 349
Bibliographical notes	p. 350
Mathematical preliminaries	p. 353
Sets	p. 353
Sequences, tuples and matrices	p. 354
Functions and relations	p. 355
Graphs	p. 356
Strings and languages	p. 357
Booleanlogic	p. 357
Probability	p. 358
Random variables	p. 359
Linear programming	p. 361
Two famous formulas	p. 365
A List of NP Optimization Problems	p. 367
Bibliography	p. 471
Index	p. 515
Table of Contents provided by Publisher. All Rights Reserved.

Complexity and Approximation

Combinatorial Optimization Problems and Their Approximability Properties

At a Glance

Hardcover

Shipping

How to return your order

Defective items

You Can Find This Book In

More in Business Applications

Microsoft 365 Excel All-in-One For Dummies

Excel for Dummies

Microsoft 365 Excel For Dummies

For Dummies (Computer/Tech)

Statistics and Data Handling for Biologists

A Student's Guide

Statistics and Data Handling for Biologists

A Student's Guide

Digital Outcasts

Moving Technology Forward without Leaving People Behind

Introduction and Applications of Machine Learning in Geotechnics

Proceedings of the 6th International Conference on Research in Management & Technovation

ICRMAT-2025

SPSS Statistics

5th Edition - A Practical Guide

Microsoft 365 Office All-in-One For Dummies

For Dummies (Computer/Tech)

MYOB For Dummies

9th Edition

Microsoft Power BI For Dummies

For Dummies (Computer/Tech)

Microsoft 365 Excel VBA Programming For Dummies

For Dummies (Computer/Tech)

Adobe Photoshop Lightroom Classic For Dummies

For Dummies (Computer/Tech)

Building Microservices

Designing Fine-Grained Systems 2nd Edition

Excel 2019 All-in-One For Dummies

All-in-One For Dummies

The Visual MBA

A Quick Guide to Everything You'll Learn in Two Years of Business School

Microsoft 365 Word For Professionals For Dummies

For Dummies (Computer/Tech)

Microsoft Power Bi Data Analyst Associate Study Guide

Prepare for the Pl-300 Exam and Apply Best Practice Design

Microsoft Planner For Dummies

Learning Generative AI Tools for Excel

Speed Up Your Everyday Tasks with Microsoft Excel, Copilot, ChatGPT, and Beyond

The Value of People Analytics

Using HR Data to Drive Business Results

Statistical Rethinking

2nd Edition - A Bayesian Course with Examples in R and STAN

Microsoft Project For Dummies

For Dummies (Computer/Tech)

Photoshop Elements 2025 For Dummies

For Dummies (Computer/Tech)

This product is categorised by