New Results in the Theory of Approximation Fast Graph Algorithms

191 Pages·2013·0.94 MB·English

by  Sushant Sachdeva

New Results in the Theory of
Approximation
Fast Graph Algorithms and Inapproximability
Sushant Sachdeva
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of
Computer Science
Advisor: Sanjeev Arora
September 2013
© Copyright by Sushant Sachdeva, 2013.
All rights reserved.
Abstract
For several basic optimization problems, it is NP-hard to find an exact solution. As
a result, understanding the best possible trade-off between the running time of an
algorithm and its approximation guarantee, is a fundamental question in theoretical
computer science, and the central goal of the theory of approximation.
There are two aspects to the theory of approximation : (1) efficient approximation
algorithms that establish trade-offs between approximation guarantee and running
time, and (2) inapproximability results that give evidence against them. In this
thesis, we contribute to both facets of the theory of approximation.
In the first part of this thesis, we present the first near-linear-time algorithm for
Balanced Separator - given a graph, partition its vertices into two roughly equal parts
without cutting too many edges - that achieves the best approximation guarantee
possible for algorithms in its class. This is a classic graph partitioning problem and
has deep connections to several areas of both theory and practice, such as metric
embeddings, Markov chains, clustering, etc.
As an important subroutine for our algorithm for Balanced Separator, we provide
a near-linear-time algorithm to simulate the heat-kernel random walk on a graph,
equivalenttocomputinge−Lv,whereListheLaplacianofthegraph, andv isavector.
This algorithm combines techniques from approximation theory and numerical linear
algebra to reduce the problem of approximating the matrix exponential to solving a
small number of Laplacian systems. We also give a reduction in the reverse direction,
from matrix inversion to matrix exponentiation, hence justifying the use of Laplacian
system solvers.
In the second part of this thesis, we prove inapproximability results for several
basic optimization problems. We address some classic scheduling problems, viz. Con-
current Open Shop and the Assembly Line problem, and variants of the Hypergraph
Vertex Cover problem. For all these problems, optimal inapproximability results were
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previously known under the Unique Games Conjecture. We are able to prove near-
optimal inapproximability results for these problems without using the conjecture.
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Acknowledgements
I owe a lot of gratitude to my advisor Sanjeev Arora. This thesis would have been
impossible without his unwavering support and guidance. His grand vision of theoret-
ical computer science, and an open-minded approach to research driven by scientific
curiosity are truly inspiring. I aspire to his simple yet crisp intuitive understanding of
complex concepts. He has helped me in innumerable ways over these years to develop
into an independent researcher.
My academic journey has been influenced positively by a number of wonderful
mentors. I would specially like to thank Nisheeth Vishnoi for his guidance. I have
greatly benefitted from our discussions, both technical, and about research and writ-
ing in general. I would also like to thank him and Microsoft Research India for
hosting me during Summer 2011. Large parts of this thesis are based on my work
with Nisheeth.
IamalsoextremelygratefultoBoazBarak, MosesCharikar, andMadhurTulsiani.
I have turned to each of them for support during several stages of my graduate
life, and have always received insightful advice for my numerous conundrums. I can
only aspire to achieve the degree of clarity their thoughts embody. I would like to
thank my undergraduate advisor, Sundar Vishwanathan, for having fostered in me
an appreciation of theory and its beauty. I wish to also thank my uncle, Suresh
Sachdeva, for having instilled in me the aspiration to be a researcher at an early age.
I wish to thank Toyota Technological Institute at Chicago for hosting me as an
internduringSummer2012, andtoYuryMakarychevforservingasmymentorduring
the stay.
ThankstoBoazBarakandBernardChazelle–Ilearnedalotaboutteachingwhile
serving as a teaching assistant for their courses. Many thanks to my entire thesis
committee: Sanjeev Arora, Mark Braverman, Moses Charikar, Bernard Chazelle, and
Nisheeth Vishnoi, for their time and their valuable suggestions.
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I have been very fortunate to have collaborated with a lot of terrific researchers
over the years and would like to thank them for everything I have learnt from them:
Sanjeev Arora, Amey Bhangale, Arnab Bhattacharyya, Julia Chuzhoy, Mark Braver-
man,RongGe,VenkatesanGuruswami,PooyaHatami,SwastikKopparty,JamesLee,
YuryMakarychev, RajsekarManokaran, MohammadMoharrami, AnkurMoitra, Ofer
Neiman, Lorenzo Orecchia, Rishi Saket, Grant Schoenebeck, Nikhil Srivastava, Mad-
hur Tulsiani, Nisheeth Vishnoi, and Pratik Worah. I would also like to acknowledge
the support of NSF grants CCF-0832797, CCF-1117309, and Sanjeev Arora’s Simons
Investigator Grant.
I would like to thank all the members of the theory groups at Princeton University
and at the Institute of Advanced Study for creating such an exciting environment for
research. Thanks also to the members of the Center for Computational Intractability
for the engaging meetings and workshops that were great opportunities to learn.
Thanks to the wonderfully helpful administrative staff at the Computer Science
department at Princeton University. Special thanks to Mitra Kelly and Melissa Law-
son for putting up with a myriad of administrative questions and requests from me.
I have shared some wonderful times with the students in theory group at the
department and would like to thank all of them for making the five years so enjoyable,
especially with all the discussions over coffee. I owe my gratitude to Aditya Bhaskara,
Moritz Hardt, Mohammad Mahmoody, Rajsekar Manokaran, Siddhartha Sen, David
Steurer, and Aravindan Vijayaraghavan, for all the advice during various stages of
graduate life. Thanks also to Rong Ge, Daniel Larkin, Shi Li, Yonatan Naamad, Huy
˜
Nguyˆen, and V. Anuradha.
Life in Princeton outside the theory group too had hardly any dull moments
due to many wonderful friends. Special thanks to my apartment mates: Sreechakra
Goparaju, for being a great companion, and for all the discussions we had on every
topic under the Sun, and Arnab Sinha, for his contagious enthusiasm for food and
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travel. Thanks also to several friends who made Princeton feel like home: Harish
B., Pathikrit Bhattacharya, Parthav Desai, Aahana Ganguly, Jaya Khanna, Radha
Kumar, Shruti Patil, Katharina Schwartz, Vinay Sitapati, V. Anuradha, and Arjun
Vijayakumar.
I have been fortunate to have some great friends nearby and would like to thank
them for several memorable weekends: Neeraja Dashaputre, Rohan Joshi, Jyotsna
Joshi, Vignesh Kasinath, Kartik Mohta, Sanika Mokashi, and Sangram Raje. I would
also like to thank several other wonderful friends, who have stayed in touch despite
the distances: Sukhada Fadnavis, Shantanu Gangal, Priya Gupta, Sonal Jain, Sudeep
Kamath, Tejas Kulkarni, Aquila Mavalankar, Neha Mhaisalkar, Aishwarya Palorath,
Soumya Venugopal, Arunanshu Roy, and Anisha Zaveri.
I would like to thank my uncle Shiv Raj Kathpalia, my aunt Lalita Kathpalia,
and my cousins Tanuj Kathpalia and Yatin Kathpalia, for being family for me in this
land far away from home.
Lastly, but most importantly, I would like to thank my family - my parents, Indra
Sachdeva and Surinder Pal Sachdeva, and my brother Prashant Sachdeva. I owe
more to their constant love, support, and understanding over the years, than words
can express. It is to them that I dedicate my thesis.
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To my family.
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Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
1 Introduction 1
1.1 Near-Linear-time Graph algorithms . . . . . . . . . . . . . . . . . . . 2
1.2 Hardness of Approximation . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Organization of this thesis . . . . . . . . . . . . . . . . . . . . . . . . 5
2 The Balanced Separator Problem and the Matrix Exponential 8
2.1 The Balanced Separator problem, and Spectral Algorithms . . . . . . 8
2.2 Approximating the Matrix Exponential . . . . . . . . . . . . . . . . . 13
2.3 Matrix Inversion is as easy as Exponentiation . . . . . . . . . . . . . 16
2.4 Uniform approximations to e−x . . . . . . . . . . . . . . . . . . . . . 18
3 Fast Spectral Algorithm for Balanced Separator 20
3.1 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Definitions and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Accelerated Heat Kernel (AHK) Random Walks . . . . . . . . . . . . 25
3.4 Algorithm Description . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5 Analysis of the Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 31
3.6 Remaining Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
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3.7 SDP Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.8 The FindCut Subroutine . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Computing exp(−A)v 48
4.1 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Lanczos Method – From Scalars to Matrices . . . . . . . . . . . . . . 53
4.3 Approximating exp(−A)v Using a Rational Approximation to e−x . 59
4.4 Beyond SDD Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5 Error Analysis for ExpRational . . . . . . . . . . . . . . . . . . . . 71
5 Matrix Inversion is as easy as Exponentiation 85
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 Integral Representation, Discretization and Smoothness . . . . . . . . 87
5.3 Proof of the Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6 Uniform Approximations to e−x 97
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2 Known Approximation Results and Discussion . . . . . . . . . . . . . 100
6.3 Proof of Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.4 Proof of Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.5 Remaining Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7 Optimal Inapproximability without UGC 112
7.1 Hardness of Approximation – A quick history . . . . . . . . . . . . . 112
7.2 Hypergraph Vertex Cover . . . . . . . . . . . . . . . . . . . . . . . . 114
7.3 Scheduling and Stochastic Vehicle Routing . . . . . . . . . . . . . . . 115
7.4 Structured Hardness for Vertex Cover . . . . . . . . . . . . . . . . . . 119
7.5 Vertex Cover on k-uniform k-partite Hypergraphs . . . . . . . . . . . 122
x