Approximation Algorithms 1st Edition by Vijay Vazirani ISBN 9783662045657 3662045656
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Authors:Vijay V. Vazirani , Series:IT & Computer [201] , Tags:Computers; Computer Science; Programming; Algorithms; Data Science; General; Business & Economics; Operations Research; Mathematics; Combinatorics; Information Technology; Discrete Mathematics , Author sort:Vazirani, Vijay V. , Ids:Google; 9783662045657 , Languages:Languages:eng , Published:Published:Mar 2013 , Publisher:Springer Science & Business Media , Comments:Comments:Although this may seem a paradox, all exact science is dominated by the idea of approximation. Bertrand Russell (1872-1970) Most natural optimization problems, including those arising in important application areas, are NP-hard. Therefore, under the widely believed con jecture that P -=/= NP, their exact solution is prohibitively time consuming. Charting the landscape of approximability of these problems, via polynomial time algorithms, therefore becomes a compelling subject of scientific inquiry in computer science and mathematics. This book presents the theory of ap proximation algorithms as it stands today. It is reasonable to expect the picture to change with time. This book is divided into three parts. In Part I we cover combinato rial algorithms for a number of important problems, using a wide variety of algorithm design techniques. The latter may give Part I a non-cohesive appearance. However, this is to be expected – nature is very rich, and we cannot expect a few tricks to help solve the diverse collection of NP-hard problems. Indeed, in this part, we have purposely refrained from tightly cat egorizing algorithmic techniques so as not to trivialize matters. Instead, we have attempted to capture, as accurately as possible, the individual character of each problem, and point out connections between problems and algorithms for solving them.
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ISBN 10: 3662045656
ISBN 13: 9783662045657
Author: Vijay Vazirani
Approximation Algorithms 1st Edition Table of contents:
1 Introduction
1.1 Lower bounding OPT
1.1.1 An approximation algorithm for cardinality vertex cover
1.1.2 Can the approximation guarantee be improved?
1.2 Well-characterized problems and min-max relations
1.3 Exercises
1.4 Notes
Part I Combinatorial Algorithms
2 Set Cover
2.1 The greedy algorithm
2.2 Layering
2.3 Application to shortest superstring
2 .4 Exercises
2.5 Notes
3 Steiner Tree and TSP
3.1 Metric Steiner tree
3.1.1 MST-based algorit
3.2.1 A simple factor 2 algorithm
3.2 Metric TSP
3.2.2 Improving the factor to 3/2
3.3 Exercises
3.4 Notes
4 Multiway Cut and k-Cut
4.1 The multiway cut problem
4.2 The minimum k-cut problem
4.3 Exercises
4.4 Notes
5 k-Center
5.1 Parametric pruning applied to metric k-center
5.2 The weighted version
5.3 Exercises
5.4 Notes
6 Feedback Vertex Set
6.1 Cyclomatic weighted graphs
6.2 Layering applied to feedback vertex set
6.3 Exercises
6.4 Notes
7 Shortest Superstring
7.1 A factor 4 algorithm
7.2 Improving to factor 3
7.2.1 Achieving half the optimal compression
7.3 Exercises
7.4 Notes
8 Knapsack
8.1 A pseudo-polynomial time algorithm for knapsack
8.2 An FPTAS for knapsack
8.3 Strong NP-hardness and the existence of FPTAS’s
8.3.1 Is an FPTAS the most desirable approximation algorithm?
8.4 Exercises
8.5 Notes
9 Bin Packing
9.1 An asymptotic PTAS
9.2 Exercises
9.3 Notes
10 Minimum Makespan Scheduling
10.1 Factor 2 algorithm
10.2 A PTAS for minimum makespan
10.2.1 Bin packing with fixed number of object sizes
10.2.2 Reducing makespan to restricted bin packing
10.3 Exercises
10.4 Notes
11 Euclidean TSP
11.1 The algorithm
11.2 Proof of correctness
11.3 Exercises
11.4 Notes
Part II LP-Based Algorithms
12 Introduction to LP-Duality
12.1 The LP-duality theorem
12.2 Min-max relations and LP-duality
12.3 Two fundamental algorithm design techniques
12.3.1 A comparison of the techniques and the notion of integrality gap
12.4 Exercises
12.5 Notes
13 Set Cover via Dual Fitting
13.1 Dual-fitting-based analysis for the greedy set cover algorithm
13.1.1 Can the approximation guarantee be improved?
13.2 Generalizations of set cover
13.2.1 Dual fitting applied to constrained set multicover
13.3 Exercises
13.4 Notes
14 Rounding Applied to Set Cover
14.1 A simple rounding algorithm
14.2 Randomized rounding
14.3 Half-integrality of vertex cover
14.4 Exercises
14.5 Notes
15 Set Cover via the Primal-Dual Schema
15.1 Overview of the schema
15.2 Primal-dual schema applied to set cover
15.3 Exercises
15.4 Notes
16 Maximum Satisfiability
16.1 Dealing with large clauses
16.2 Derandomizing via the method of conditional expectation
16.3 Dealing with small clauses via LP-rounding
16.4 A 3/4 factor algorithm
16.5 Exercises
16.6 Notes
17 Scheduling on Unrelated Parallel Machines
17.1 Parametric pruning in an LP setting
17.2 Properties of extreme point solutions
17.3 The algorithm
17.4 Additional properties of extreme point solutions
17.5 Exercises
17.6 Notes
18 Multicut and Integer Multicommodity Flow in Trees
18.1 The problems and their LP-relaxations
18.2 Primal-dual schema based algorithm
18.3 Exercises
18.4 Notes
19 Multiway Cut
19.1 An interesting LP-relaxation
19.2 Randomized rounding algorithm
19.3 Half-integrality of node multiway cut
19.4 Exercises
19.5 Notes
20 Multicut in General Graphs
20.1 Sum multicommodity flow
20.2 LP-rounding-based algorithm
20.2.1 Growing a region: the continuous process
20.2.2 The discrete process
20.2.3 Finding successive regions
20.3 A tight example
20.4 Some applications of multicut
20.5 Exercises
20.6 Notes
21 Sparsest Cut
21.1 Demands multicommodity flow
21.2 Linear programming formulation
21.3 Metrics, cut packings, and L 1 -embeddability
21.3.1 Cut packings for metrics
21.3.2 L 1 -embeddability of metrics
21.4 Low distortion £1-embeddings for metrics
21.4.1 Ensuring that a single edge is not overshrunk
21.4.2 Ensuring that no edge is overshrunk
21.5 LP-rounding-based algorithm
21.6 Applications
21.6.1 Edge expansion
21.6.2 Conductance
21.6.3 Balanced cut
21.6.4 Minimum cut linear arrangement
21.7 Exercises
21.8 Notes
22 Steiner Forest
22.1 LP-relaxation and dual
22.2 Primal-dual schema with synchronization
22.3 Analysis
22.4 Exercises
22.5 Notes
23 Steiner Network
23.1 LP-relaxation and half-integrality
23.2 The technique of iterated rounding
23.3 Characterizing extreme point solutions
23.4 A counting argument
23.5 Exercises
23.6 Notes
24 Facility Location
24.1 An intuitive understanding of the dual
24.2 Relaxing primal complementary slackness conditions
24.3 Primal-dual schema based algorithm
24.4 Analysis
24.4.1 Running time
24.4.2 Tight example
24.5 Exercises
24.6 Notes
25 k-Median
25.1 LP-relaxation and dual
25.2 The high-level idea
25.3 Randomized rounding
25.3.1 Derandomization
25.3.2 Running time
25.3.3 Tight example
25.3.4 Integrality gap
25.4 A Lagrangian relaxation technique for approximation algorithms
25.5 Exercises
25.6 Notes
26 Semidefinite Programming
26.1 Strict quadratic programs and vector programs
26.2 Properties of positive semidefinite matrices
26.3 The semidefinite programming problem
26.4 Randomized rounding algorithm
26.5 Improving the guarantee for MAX-2SAT
26.6 Exercises
26.7 Notes
Part III Other Topics
27 Shortest Vector
27.1 Bases, determinants, and orthogonality defect
27.2 The algorithms of Euclid and Gauss
27.3 Lower bounding OPT using Gram-Schmidt orthogonalization
27.4 Extension to n dimensions
27.5 The dual lattice and its algorithmic use
27.6 Exercises
27.7 Notes
28 Counting Problems
28.1 Counting DNF solutions
28.2 Network reliability
28.2.1 Upperbounding the number of near-minimum cuts
28.2.2 Analysis
28.3 Exercises
28.4 Notes
29 Hardness of Approximation
29.1 Reductions, gaps, and hardness factors
29.2 The PCP theorem
29.3 Hardness of MAX-3SAT
29.4 Hardness of MAX-3SAT with bounded occurrence of variables
29.5 Hardness of vertex cover and Steiner tree
29.6 Hardness of clique
29.7 Hardness of set cover
29. 7.1 The two-prover one-round characterization of NP
29.7.2 The gadget
29. 7.3 Reducing error probability by parallel repetition
29.7.4 The reduction
29.8 Exercises
29.9 Notes
30 Open Problems
30.1 Problems having constant factor algorithms
30.2 Other optimization problems
30.3 Counting problems
30.4 Notes
A An Overview of Complexity Theory for the Algorithm Designer
B Basic Facts from Probability Theory
B.1 Expectation and moments
B.2 Deviations from the mean
B.3 Basic distributions
B.4 Notes
References
Problem Index
Subject Index
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