The Design of Approximation Algorithms

Transcription

1 The Design of Approximation Algorithms David P. Williamson Cornell University David B. Shmoys Cornell University m Щ0 CAMBRIDGE UNIVERSITY PRESS

2 Contents Preface page ix I An Introduction to the Techniques 1 An Introduction to Approximation Algorithms The Whats and Whys of Approximation Algorithms An Introduction to the Techniques and to Linear Programming: The Set Cover Problem A Deterministic Rounding Algorithm Rounding a Dual Solution Constructing a Dual Solution: The Primal-Dual Method A Greedy Algorithm A Randomized Rounding Algorithm 19 Exercises 22 Chapter Notes 24 2 Greedy Algorithms and Local Search Scheduling Jobs with Deadlines on a Single Machine Thefc-Center Problem Scheduling Jobs on Identical Parallel Machines The Traveling Salesman Problem Maximizing Float in Bank Accounts Finding Minimum-Degree Spanning Trees Edge Coloring 47 Exercises 51 Chapter Notes 54 3 Rounding Data and Dynamic Programming The Knapsack Problem Scheduling Jobs on Identical Parallel Machines 61 v

3 vi CONTENTS 3.3 The Bin-Packing Problem 66 Exercises 70 Chapter Notes 71 4 Deterministic Rounding of Linear Programs Minimizing the Sum of Completion Times on a Single Machine Minimizing the Weighted Sum of Completion Times on a Single Machine Solving Large Linear Programs in Polynomial Time via the Ellipsoid Method The Prize-Collecting Steiner Tree Problem The Uncapacitated Facility Location Problem The Bin-Packing Problem 88 Exercises 93 Chapter Notes 96 5 Random Sampling and Randomized Rounding of Linear Programs Simple Algorithms for MAX SAT and MAX CUT Derandomization Flipping Biased Coins Randomized Rounding Choosing the Better of Two Solutions Nonlinear Randomized Rounding The Prize-Collecting Steiner Tree Problem The Uncapacitated Facility Location Problem Scheduling a Single Machine with Release Dates Chernoff Bounds Integer Multicommodity Flows Random Sampling and Coloring Dense 3-Colorable Graphs 130 Exercises 132 Chapter Notes Randomized Rounding of Semideflnite Programs A Brief Introduction to Semideflnite Programming Finding Large Cuts Approximating Quadratic Programs Finding a Correlation Clustering Coloring 3-Colorable Graphs 149 Exercises 152 Chapter Notes The Primal-Dual Method The Set Cover Problem: A Review Choosing Variables to Increase: The Feedback Vertex Set Problem in Undirected Graphs Cleaning Up the Primal Solution: The Shortest s-t Path Problem Increasing Multiple Variables at Once: The Generalized Steiner Tree Problem 167

4 CONTENTS vii 7.5 Strengthening Inequalities: The Minimum Knapsack Problem The Uncapacitated Facility Location Problem Lagrangean Relaxation and thefc-medianproblem 182 Exercises 188 Chapter Notes Cuts and Metrics The Multiway Cut Problem and a Minimum-Cut-Based Algorithm The Multiway Cut Problem and an LP Rounding Algorithm The Multicut Problem Balanced Cuts Probabilistic Approximation of Metrics by Tree Metrics An Application of Tree Metrics: Buy-at-Bulk Network Design Spreading Metrics, Tree Metrics, and Linear Arrangement 218 Exercises 224 Chapter Notes 227 II Further Uses of the Techniques 9 Further Uses of Greedy and Local Search Algorithms A Local Search Algorithm for the Uncapacitated Facility Location Problem A Local Search Algorithm for the ^-Median Problem Minimum-Degree Spanning Trees A Greedy Algorithm for the Uncapacitated Facility Location Problem 246 Exercises 251 Chapter Notes Further Uses of Rounding Data and Dynamic Programming The Euclidean Traveling Salesman Problem The Maximum Independent Set Problem in Planar Graphs 267 Exercises 275 Chapter Notes Further Uses of Deterministic Rounding of Linear Programs The Generalized Assignment Problem Minimum-Cost Bounded-Degree Spanning Trees Survivable Network Design and Iterated Rounding 296 Exercises 303 Chapter Notes Further Uses of Random Sampling and Randomized Rounding of Linear Programs The Uncapacitated Facility Location Problem The Single-Source Rent-or-Buy Problem The Steiner Tree Problem 316

5 viii CONTENTS 12.4 Everything at Once: Finding a Large Cut in a Dense Graph 324 Exercises 329 Chapter Notes Further Uses of Randomized Rounding of Semideflnite Programs Approximating Quadratic Programs Coloring 3-Colorable Graphs Unique Games 345 Exercises 352 Chapter Notes Further Uses of the Primal-Dual Method The Prize-Collecting Steiner Tree Problem The Feedback Vertex Set Problem in Undirected Graphs 360 Exercises 366 Chapter Notes Further Uses of Cuts and Metrics Low-Distortion Embeddings and the Sparsest Cut Problem Oblivious Routing and Cut-Tree Packings Cut-Tree Packings and the Minimum Bisection Problem The Uniform Sparsest Cut Problem 387 Exercises 406 Chapter Notes Techniques in Proving the Hardness of Approximation Reductions from NP-Complete Problems Reductions that Preserve Approximation Reductions from Probabilistically Checkable Proofs Reductions from Label Cover Reductions from Unique Games 442 Chapter Notes Open Problems 453 Appendix A: Linear Programming 459 Appendix B: NP-Completeness 465 Bibliography 469 Author Index 485 Subject Index 491