Table Of ContentProperty Testing and Combinatorial
Approximation
8
0
0
2
-
9
0
-
8
0 Arie Matsliah
0
2
-
D
H
P
s
i
s
e
h
T
.
D
.
h
P
-
t
n
e
m
t
r
a
p
e
D
e
c
n
e
i
c
S
r
e
t
u
p
m
o
C
-
n
o
i
n
h
c
e
T
8
0
0
2
-
9
0
-
8
0
0
2
-
D
H
P
s
i
s
e
h
T
.
D
.
h
P
-
t
n
e
m
t
r
a
p
e
D
e
c
n
e
i
c
S
r
e
t
u
p
m
o
C
-
n
o
i
n
h
c
e
T
Property Testing and Combinatorial
Approximation
8
0
0
2
-
9
0
-
8
0 Research Thesis
0
2
-
D
H
P
s
i
s
e
h Submitted in Partial Fulflllment of the Requirements
T for the Degree of Doctor of Philosophy
.
D
.
h
P
-
t
n
e
m
t
r
a
p
e
D Arie Matsliah
e
c
n
e
i
c
S
r
e
t
u
p
m Submitted to the Senate of the
o Technion - Israel Institute of Technology
C
-
n
o
i
n
h
c
e
T
Tamuz 5768 Haifa July 2008
8
0
0
2
-
9
0
-
8
0
0
2
-
D
H
P
s
i
s
e
h
T
.
D
.
h
P
-
t
n
e
m
t
r
a
p
e
D
e
c
n
e
i
c
S
r
e
t
u
p
m
o
C
-
n
o
i
n
h
c
e
T
The research thesis was done under the supervision of
Assoc. Prof. Eldar Fischer
in the Department of Computer Science.
8
0
0
2
-
9
0
-
8
0
0
2
-
D
H
P
Acknowledgements
s
i
s
e
h
T
I am grateful to my supervisor - Eldar Fischer, for guiding me devotedly for the
.
D last flve years, for his continued encouragement and for many invaluable suggestions
h. in this work. I would also like to include my gratitude to Eli Ben-Sasson, especially
P
for his willingness to teach and assist in any given moment. Furthermore, I am
- deeply indebted to my collaborators in this work: Sourav Chakraborty, Prahladh
t
n Harsha, Oded Lachish, Ilan Newman, Asaf Shapira and Orly Yahalom.
e
m
t
r
a
p
e
D
e
c
n
e
i
c
S
r
e
t
u
p
m
o
C
-
n
o
i
n
h
c
e
T
The generous flnancial help of the Technion is gratefully acknowledged
8
0
0
2
-
9
0
-
8
0
0
2
-
D
H
P
s
i
s
e
h
T
.
D
.
h
P
-
t
n
e
m
t
r
a
p
e
D
e
c
n
e
i
c
S
r
e
t
u
p
m
o
C
-
n
o
i
n
h
c
e
T
8
0
0
2
Contents
-
9
0
-
8
0
0
2
D- List of Figures v
H
P Abstract 1
s
si 1 General introduction 3
e
h 1.1 Combinatorial property testing . . . . . . . . . . . . . . . . . . . . . . . . . 3
T
1.1.1 Property testing in the dense graph (and hypergraph) model . . . . 4
.
D
1.1.2 Testing graph properties in other models . . . . . . . . . . . . . . . 5
.
h
P 1.1.3 Testing massively parameterized properties . . . . . . . . . . . . . . 5
- 1.2 Probabilistically checkable proofs of proximity - PCPPs . . . . . . . . . . . 6
t
n
e
2 Summary of results 8
m
t 2.1 Testing graph isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
r
a
p 2.2 Approximate hypergraph partitioning and applications . . . . . . . . . . . . 8
e
D 2.3 Testing st-connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
e 2.4 Testing orientations for being Eulerian . . . . . . . . . . . . . . . . . . . . . 9
c
n 2.5 Length-Soundness tradeofis for 3-query PCPPs . . . . . . . . . . . . . . . . 10
e
i
c
S 3 Global deflnitions and preliminaries 11
r 3.1 General notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
e
t
u 3.2 Property testers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
p
m 3.3 Graphs and graph properties . . . . . . . . . . . . . . . . . . . . . . . . . . 12
o 3.3.1 Distance between graphs and properties . . . . . . . . . . . . . . . . 12
C
3.4 Restriction, variation distance and Yao’s method . . . . . . . . . . . . . . . 13
-
n
o
i
n
I Dense Graphs and Hypergraphs 14
h
c
e
T 4 Testing graph isomorphism 15
4.1 Background and introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Speciflc deflnitions and preliminaries . . . . . . . . . . . . . . . . . . . . . . 17
i
4.3 One-sided testers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3.1 One-sided testing of two unknown graphs . . . . . . . . . . . . . . . 19
4.3.2 One-sided testing where one of the graphs is known in advance . . . 23
8
4.4 Two-sided testers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
0
0 4.4.1 Two-sided testing where one of the graphs is known in advance . . . 24
2
- 4.4.2 Two-sided testing of two unknown graphs . . . . . . . . . . . . . . . 32
9
4.5 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
0
-
8
0 5 Approximate hypergraph partitioning and applications 37
0
2 5.1 Background and introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 38
-
D 5.2 Extension of the GGR-algorithm . . . . . . . . . . . . . . . . . . . . . . . . 39
H
5.2.1 The partition property . . . . . . . . . . . . . . . . . . . . . . . . . . 40
P
5.3 Immediate applications of the theorems . . . . . . . . . . . . . . . . . . . . 42
s
i
s 5.3.1 Estimating the maximum number of satisflable clauses in a k-CNF . 43
e
h 5.4 Finding a regular partition of a graph . . . . . . . . . . . . . . . . . . . . . 44
T
5.4.1 Background and statement . . . . . . . . . . . . . . . . . . . . . . . 44
.
D
5.4.2 The proof idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
.
h
P 5.4.3 Proof of Theorem 5.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . 48
- 5.5 Regular partition of a hypergraph. . . . . . . . . . . . . . . . . . . . . . . . 52
t
n 5.5.1 Proof for Theorem 5.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . 53
e
m 5.6 Overview of the proof of Theorems 5.2.1 and 5.2.2 . . . . . . . . . . . . . . 56
t
r 5.7 The proof of Theorem 5.2.1 and Theorem 5.2.2 . . . . . . . . . . . . . . . . 57
a
p
5.7.1 Proof of Lemma 5.7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 60
e
D
5.7.2 Algorithm A - proof of Lemma 5.7.4 . . . . . . . . . . . . . . . . 61
1="
e
5.7.3 Some possible ad-hoc optimizations for properties . . . . . . . . . . 67
c
n
e 5.7.4 The last missing details of the proof . . . . . . . . . . . . . . . . . . 67
i
c 5.8 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
S
r
e
t
u II Massively Parameterized Properties 72
p
m
o 6 Introduction 73
C
6.1 The orientation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
-
n 6.2 Speciflc deflnitions and preliminaries . . . . . . . . . . . . . . . . . . . . . . 74
o
i 6.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
n
h 6.2.2 Orientation distance, properties and testers . . . . . . . . . . . . . . 75
c
e
T
7 Testing st-connectivity 76
7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.1.1 Connectivity programs and Branching programs . . . . . . . . . . . 78
ii
7.2 The main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.2.1 Proof overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.3 Reducing general graphs to connectivity programs . . . . . . . . . . . . . . 81
8
7.3.1 Reducibility between st-connectivity instances . . . . . . . . . . . . 81
0
0 7.3.2 Reduction to graphs having high-diameter subgraphs . . . . . . . . . 81
2
- 7.3.3 Properties of "-long graphs . . . . . . . . . . . . . . . . . . . . . . . 83
9
7.3.4 Reduction to bounded width graphs . . . . . . . . . . . . . . . . . . 83
0
-
8 7.3.5 Reducing bounded width graphs to st-connectivity programs . . . . 84
0
0 7.4 Reducing st-connectivity programs to branching programs . . . . . . . . . . 84
2
- 7.4.1 Converting clustered branching programs to non-clustered ones . . . 86
D
7.5 Wrapping up { proof of Theorem 7.2.1 . . . . . . . . . . . . . . . . . . . . . 87
H
P 7.6 Completing the missing proofs . . . . . . . . . . . . . . . . . . . . . . . . . 88
s 7.6.1 Proof of Lemma 7.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 88
i
s
e 7.6.2 Proof of Lemma 7.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 90
h
T
8 Testing for Eulerian orientations 92
.
D
8.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
.
h
P 8.2 Speciflc preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
- 8.3 Lower bounds for bounded-degree graphs . . . . . . . . . . . . . . . . . . . 95
t
n 8.3.1 Overview of the proof of Theorem 8.3.1 . . . . . . . . . . . . . . . . 96
e
m 8.3.2 Torus { formal deflnition . . . . . . . . . . . . . . . . . . . . . . . . 96
t
r 8.3.3 Deflning two auxiliary distributions . . . . . . . . . . . . . . . . . . 97
a
p
8.3.4 Deflning the distributions and . . . . . . . . . . . . . . . . . 99
e m m
P F
D
8.3.5 Bounding the probability of distinguishing between and . . . 102
m m
P F
e
8.3.6 Proof of Theorem 8.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . 104
c
n
e 8.3.7 Overview of the proof of Theorem 8.3.2 . . . . . . . . . . . . . . . . 104
i
c 8.3.8 Proof of Theorem 8.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . 105
S
r
e
t
u III Probabilistically Checkable Proofs of Proximity 109
p
m
o 9 Length-Soundness tradeofis for 3-query PCPPs 110
C
9.1 Background and introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 110
-
n 9.1.1 Informal description of the results . . . . . . . . . . . . . . . . . . . 111
o
i 9.1.2 Context and motivation . . . . . . . . . . . . . . . . . . . . . . . . . 113
n
h 9.1.3 Proof techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
c
e 9.2 Speciflc deflnitions and statement of the main results . . . . . . . . . . . . . 117
T
9.2.1 Probabilistically checkable proofs of proximity (PCPPs) . . . . . . . 117
9.2.2 Soundness deflciency . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
iii
9.2.3 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.2.4 Inspective PCPPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
9.3 Long PCPPs with best possible soundness . . . . . . . . . . . . . . . . . . . 123
8
9.3.1 Fourier transform { preliminaries . . . . . . . . . . . . . . . . . . . . 123
0
0 9.3.2 Proof of Theorem 9.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 124
2
- 9.4 Proof of Length-Soundness tradeofi (Theorem 9.2.3) . . . . . . . . . . . . . 127
9
9.4.1 Constraint graphs and the generalized Decomposition Lemma . . . . 127
0
-
8 9.4.2 The uniform (sparse) verifler lemma . . . . . . . . . . . . . . . . . . 130
0
0 9.4.3 Best soundness for inspective veriflers (proof of Theorem 9.2.8) . . . 132
2
- 9.4.4 Proof of Theorem 9.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . 134
D
9.4.5 Proof of Lemma 9.4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 134
H
P 9.5 Proof of Length-Soundness tradeofi for linear veriflers (Theorem 9.2.9) . . . 137
s 9.5.1 Proof of Theorem 9.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . 138
i
s
e 9.5.2 The Decomposition Lemma . . . . . . . . . . . . . . . . . . . . . . . 140
h
T 9.5.3 Proof of Lemma 9.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 141
.
D
.
h IV Bibliography 148
P
-
t
n
e
m
t
r
a
p
e
D
e
c
n
e
i
c
S
r
e
t
u
p
m
o
C
-
n
o
i
n
h
c
e
T
iv
