Approximating NP-hard Problems Efficient Algorithms and their Limits

314 Pages·2009·2.41 MB·English

Approximating NP-hard Problems
Efficient Algorithms and their Limits
Prasad Raghavendra
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
University of Washington
2009
Program Authorized to Offer Degree: Computer Science and Engineering

University of Washington
Graduate School
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Prasad Raghavendra
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and that any and all revisions required by the final
examining committee have been made.
Chair of the Supervisory Committee:
Venkatesan Guruswami
Reading Committee:
Venkatesan Guruswami
Paul Beame
James R Lee
Date:

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University of Washington
Abstract
Approximating NP-hard Problems
Efficient Algorithms and their Limits
Prasad Raghavendra
Chair of the Supervisory Committee:
Associate Professor Venkatesan Guruswami
Computer Science and Engineering
Most combinatorial optimization problems are NP-hard to solve optimally. A natural
approach to cope with this intractability is to design an “approximation algorithm” – an
efficient algorithm that is guaranteed to produce a good approximation to the optimum
solution. The last two decades has witnessed tremendous developments in the design of
approximation algorithms mostly fueled by convex optimization techniques such as linear
or semidefinite programming.
Inthisthesis,wepresentalgorithmicandlowerboundresultsthatcharacterizethepower
and limitations of these techniques on large classes of combinatorial optimization problems.
The thesis identifies a specific simple semidefinite program and demonstrates the following:
– Thissemidefiniteprogramyieldstheoptimalapproximationtoeveryprobleminoneof
the large classes such as constraint satisfaction problems (CSP), metric labeling prob-
lems and ordering constraint satisfaction problems under the Unique Games Con-
jecture (UGC). To show this, we exhibit a general black-box reduction from hard
instances to a linear/semidefinite program to correspondinghardness results based on
the UGC. Not only does this confirm a widely suspected connection, it settles the
approximability of classic optimization problems such as CSPs, Multiway Cut and
Maximum Acyclic Subgraph under UGC.
– The thesis presents a generic algorithm for constraint satisfaction problems (CSP)
based on this semidefinite program. Irrespective of the truth of UGC, this generic
algorithm is guaranteed to obtain an approximation at least as good as all known
algorithms for specific CSPs.
– Independent of the truth of UGC, the approximation obtained by this semidefinite
program cannot be improved by any convex relaxation that is obtained by including
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any valid constraints on at most O(2(loglogN)4) vectors.

TABLE OF CONTENTS
Page
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Relaxation and Rounding Methodology . . . . . . . . . . . . . . . . . . . . . 5
1.3 Relaxation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Understanding the Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Integrality Gaps vs Hardness Results . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Brief Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 11
Chapter 2: Preliminaries And Organization of Thesis . . . . . . . . . . . . . . . . 13
2.1 Relaxation and Rounding: Examples . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Definitions and Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Problem Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Generalized Constraint Satisfaction Problems . . . . . . . . . . . . . . . . . . 22
2.5 Label Cover and Unique Games . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6 Results and Organization of thesis . . . . . . . . . . . . . . . . . . . . . . . . 25
Chapter 3: Mathematical Tools. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Probability Spaces and Random Variables . . . . . . . . . . . . . . . . . . . . 31
3.3 Harmonic Analysis of Boolean Functions . . . . . . . . . . . . . . . . . . . . . 32
3.4 Functions on Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5 Gaussian Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.6 Invariance Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.7 Noise Stability Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
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Part I: Algorithmic Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Chapter 4: Linear and Semidefinite Programming Relaxations . . . . . . . . . . . 49
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Comparing Relaxations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3 Local Distributions and Consistency . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 A Simple LP Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.5 A Simple SDP Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.6 Comparison with Relaxations in Literature . . . . . . . . . . . . . . . . . . . 59
4.7 Stronger Relaxations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.8 Robustness and Smoothing of the LC relaxation . . . . . . . . . . . . . . . . . 65
4.9 Robustness of LH and SA relaxations . . . . . . . . . . . . . . . . . . . . . . 70
r r
Chapter 5: A Generic Rounding Scheme. . . . . . . . . . . . . . . . . . . . . . . . 77
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3 Proof Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.5 Rounding General CSPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Part II: The Unique Games Barrier . . . . . . . . . . . . . . . . . . . . . . . . 88
Chapter 6: Dictatorship Tests, Rounding Schemes and Unique Games Conjecture 89
6.1 Dictatorship Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 Emerging Connections (history) . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.3 From Dictatorship Tests to UG-hardness Results . . . . . . . . . . . . . . . . 94
6.4 From Integrality Gaps to Dictatorship Tests . . . . . . . . . . . . . . . . . . . 98
6.5 Formal Proof of Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.6 Dictatorship Tests and Rounding Schemes . . . . . . . . . . . . . . . . . . . . 108
6.7 From UG-hardness to Integrality Gaps . . . . . . . . . . . . . . . . . . . . . . 112
6.8 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Chapter 7: General Constraint Satisfaction Problems . . . . . . . . . . . . . . . . 119
7.1 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.3 From Integrality Gaps to Dictatorship Tests . . . . . . . . . . . . . . . . . . . 126
7.4 Soundness of Dictatorship Test DICTε . . . . . . . . . . . . . . . . . . . . . 128
V,µ
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