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首页组合优化:精确算法和近似算法Combinatorial Optimization: Exact and Approximate Algorithms
组合优化:精确算法和近似算法Combinatorial Optimization: Exact and Approximate ...
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斯坦福大学教授的CS261类:优化和算法范例的讲义。 它们涵盖了近似算法,精确优化和在线算法的主题。
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Combinatorial Optimization:
Exact and Approximate Algorithms
Luca Trevisan
Stanford University
March 19, 2011
Foreword
These are minimally edited lecture notes from the class CS261: Optimization and Algorith-
mic Paradigms that I taught at Stanford in the Winter 2011 term. The following 18 lectures
cover topics in approximation algorithms, exact optimization, and online algorithms.
I gratefully acknowledge the support of the National Science Foundation, under grant CCF
1017403. Any opinions, findings and conclusions or recommendations expressed in these
notes are my own and do not necessarily reflect the views of the National Science Foundation.
Luca Trevisan, San Francisco, March 19, 2011.
c
2011 by Luca Trevisan
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs
3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/
licenses/by-nc-nd/3.0/ or send a letter to Creative Commons, 171 Second Street, Suite
300, San Francisco, California, 94105, USA.
i
ii
Contents
Foreword i
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Vertex Cover Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Steiner Tree Approximation 7
2.1 Approximating the Metric Steiner Tree Problem . . . . . . . . . . . . . . . 7
2.2 Metric versus General Steiner Tree . . . . . . . . . . . . . . . . . . . . . . . 10
3 TSP and Eulerian Cycles 13
3.1 The Traveling Salesman Problem . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 A 2-approximate Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Eulerian Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 Eulerian Cycles and TSP Approximation . . . . . . . . . . . . . . . . . . . 19
4 TSP and Set Cover 21
4.1 Better Approximation of the Traveling Salesman Problem . . . . . . . . . . 21
4.2 The Set Cover Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 Set Cover versus Vertex Cover . . . . . . . . . . . . . . . . . . . . . . . . . 29
5 Linear Programming 31
5.1 Introduction to Linear Programming . . . . . . . . . . . . . . . . . . . . . . 31
5.2 A Geometric Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.2.1 A 2-Dimensional Example . . . . . . . . . . . . . . . . . . . . . . . . 32
iii
iv CONTENTS
5.2.2 A 3-Dimensional Example . . . . . . . . . . . . . . . . . . . . . . . . 34
5.2.3 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.2.4 Polynomial Time Algorithms for LInear Programming . . . . . . . . 37
5.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3 Standard Form for Linear Programs . . . . . . . . . . . . . . . . . . . . . . 39
6 Linear Programming Duality 41
6.1 The Dual of a Linear Program . . . . . . . . . . . . . . . . . . . . . . . . . 41
7 Rounding Linear Programs 47
7.1 Linear Programming Relaxations . . . . . . . . . . . . . . . . . . . . . . . . 47
7.2 The Weighted Vertex Cover Problem . . . . . . . . . . . . . . . . . . . . . . 48
7.3 A Linear Programming Relaxation of Vertex Cover . . . . . . . . . . . . . . 50
7.4 The Dual of the LP Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 51
7.5 Linear-Time 2-Approximation of Weighted Vertex Cover . . . . . . . . . . . 52
8 Randomized Rounding 57
8.1 A Linear Programming Relaxation of Set Cover . . . . . . . . . . . . . . . . 57
8.2 The Dual of the Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
9 Max Flow 65
9.1 Flows in Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
10 The Fattest Path 73
10.1 The “fattest” augmenting path heuristic . . . . . . . . . . . . . . . . . . . . 74
10.1.1 Dijkstra’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
10.1.2 Adaptation to find a fattest path . . . . . . . . . . . . . . . . . . . . 76
10.1.3 Analysis of the fattest augmenting path heuristic . . . . . . . . . . . 77
11 Strongly Polynomial Time Algorithms 79
11.1 Flow Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
11.2 The Edmonds-Karp Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 81
12 The Push-Relabel Algorithm 83
12.1 The Push-Relabel Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
12.2 Analysis of the Push-Relabel Algorithm . . . . . . . . . . . . . . . . . . . . 85
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