Tali Kaufman Joint work with Irit Dinur Codes as CSPs A constraint satisfaction problem CSP is a sequence of constraints over variables f 1 x 1 x 2 x 5 f 2 x 5 x 1 x ID: 279450
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Slide1
Locally Testable Codes and Expanders
Tali Kaufman
Joint work with Irit DinurSlide2
Codes as CSPs
A constraint satisfaction problem (CSP) is a sequence of constraints over variables:
f
1
(x
1
,x
2
,x
5
), f
2
(x
5
,x
1
,x
15
), …
For example, max-3SAT, max-3col, …
CSP describes a code if every two satisfying assignments are far apart (in Hamming distance). The minimum distance between a pair of sat assignments is the
distance
of the code.
Today : CSPs that yield codes.
Challenge: CSPs that yield large and robust codes.Slide3
Robust Codes
CSP describes a code if every two satisfying assignments are far apart (in Hamming distance).
CSP describes a
robust code
if a vector that is far from the code falsifies many constraints.
Robust codes are called LTCs (strong relation to PCP).
Want to understand: What makes a code robust? Slide4
Structure
One can associate a “constraint hyper-graph” with a CSP by placing a vertex per variable, and a hyper-edge per clause.
Two “separate” aspects of a CSP are
The graph structure.
The constraint type : what functions f
i
do we allow.
Today we focus on the structure.Slide5
Constraint (Hyper graph)
f
1
(x
1
,x
2
,x
3
), f2 (x3,x4,x5), …
x
1
x
3
x
4
x5
x2
x
6Slide6
Constraint graph
f
1
(x
1
,x
2
,x
3
), f2 (x3,x4,x5), …
We replace every hyper-edge with a clique between its vertices, so that we deal with graphs instead of hyper-graphs.
x
1
x
3
x
4
x5
x
2
x
6
x
1
x
3
x
4
x
5
x
2
x
6Slide7
Locally Testable Codes (LTCs)
An LTC is given by a constraint graph G on n vertices.
The valid codewords C(G)
⊆{0,1}
n
are all assignments to [n] that satisfy all constraints.
Two required properties of the constraint graph:
1. Distance
: Every two codewords differ on
Ω
(n) coordinates. The minimum distance is the code distance.2. Robustness: Assignment that is γ-far (in Hamming distance) from the code, falsifies at least ργ fraction of the constraints (for some fixed
ρ > 0 ).Slide8
Our question(s) for today
What is the relation between the structure of the constraint graph and the code obtained.
What graphs lead to codes?
[
Sipser-Spielman
]
What graphs lead to robust codes (LTCs)?Slide9
Motivation
How can we construct LTCs?
Understand whether good LTCs exist.
Construct new, simpler LTCs, PCPs.
Understand the structure of general CSPs; CSPs that are hard to approximate. Slide10
What graphs lead to codes
[
Sipser-Spielman
]: Expanders!
Expanders are graphs without small cuts.
G=(V,E) is an r-expander if for every
S
⊆
V, |S|
≤ |V|/2 E(S,V-S) > r |S|
SV-S
In good expanders small setsexpands even more.Slide11
Expanders lead to codes - The Idea
Constraint graph: good expander.
Constraints: local view; all zeros, or many non zeros.
Satisfying assignment must have a distance.
x
4
x
5
x
9
x
7x2
x
15
x1
x
3
x10
x
6
x
8
x
12
x
11
x
14
x
13
Non zero variables
zero neighborsSlide12
Satisfying assignment can not have low support:
- Small set of non-zeros: by expansion its zero neighborhood is large.
- Contradicts the requirement that each local constraint assigned many non zeros.
x
4
x
5
x
9
x
7
x2
x
15
x1
x3
x
10
x
6
x
8
x
12
x
11
x
14
x
13
Non zero variables
zero neighbors
Expanders lead to codes - The Idea
Other issue:
How to ensure large code?Slide13
What graphs lead to robust
codes?
[
Ben-Sasson, Harsha, Raskhodnikova
]: Very good expanders yield codes that are NOT robust/LTC.
Idea:
Remove one constraint from the constraint graph.
The resulted graph is still an expander.
All satisfying assignments of the new graph are far apart. Some sat assignments of the new graph do not sat the original graph.Conclusion: there is a vector far from code that falsifies only one constraint from the constraint graph.Slide14
What graphs lead to robust
codes?
[
BHR
]: Very good expanders yield codes that are not robust/LTC.
There is a vector far from code that falsifies only one constraint from the constraint graph.
But: Maybe can test this code using *other* short constraint. I.e. constraints that are
implied
by original constraints.
Main point: Due to the high expansion, there are no short constraints that are implied by the original short constraints.Example: f1 (x1,x2,x3
): x1 + x2 + x3 = 0f2 (x4,x5,x6): x4 + x5 + x6 = 0
f3
(x7,x8,x9): x7 + x
8 + x9 = 0f4 (x3,x
6,x9): x3 + x6 + x
9 = 0f5 (x2,x5,x
8): x2 + x5 + x8 = 0
Imply:
f
6
(x
1
,x
4
,x
7
): x
1
+ x
4
+ x7 = 0Slide15
Our Conclusions so far
High expansion of the constraint graph implies code (distance).
High expansion implies NON robustness.
What can we say about the structure of the constraint graphs of LTCs?Slide16
Structure theorem for LTCs
C is
ρ
-LTC : w
γ
-away from C rejected with probability >
ρ γ
Theorem [Dinur-K]
: The constraint graph of an LTC is a small set expander; all sets up to linear size expands. Theorem: In ρ-LTC with r. distance
Δ and with constraint graph G=(V,E), every S ⊆ V ; |S|≤ 0.75Δ|V| expands: E(S,V-S) > ρ
d/3
|S| d = |E|/|V|.Slide17
LTC decomposes on sparse cuts
C is
ρ
-LTC
: vector
γ
-away from C rejected with probability >
ρ γ
Decomposition on sparse cuts: Let |S| = γ|V|. If E(S,V-S) is a sparse cut, i.e., E(S,V-S) < ρ
d/3|S|= ργ/3|E| then C =~ γ/3 CS
x C
V-S = {xSyV-S
| xS
C|S , yV-S
C|V-S}
Note C
⊆
C
S
x C
V-S
Every c
C
S
x C
V-S
is
γ
/3-close to some codeword from C.
Main point: Such c can be rejected only by edges (constraints) that cross the cut (S,V-S) and since the cut is sparse, there are too few of those edges.Slide18
Sparse cuts occur only on large sets
C is
ρ
-LTC
: vector
γ
-away from C rejected with probability >
ρ γ
E(S,V-S) is a sparse cut: |S| = γ|V|= γ n , E(S,V-S) < ρd/3 |S|= ργ/3 |E|
C =~ γ/3 CS x CV-S
Pick a ≠ a’ in CS
with distance(a,a’) > |S|/3 = γ/3 n :
Consider aS bV-S
C ; a’S bV-S C
S x CV-S Distance of a’b from C is < γ
/3n since C=~ γ/3
C
S
x C
V-S
The closest codeword of C to
a’
S
b
V-S
is w
≠
a
S
b
V-S
By triangle-inequality distance (a’b,ab) = distance (a,a’) > (Δ - γ
/3) nIn particular |S|= γ n > (Δ
- γ/3) n ; |S| > 3/4 Δ
na’b
ab C
> γ/3 n
w
C
<
γ
/3
n by LTC
>
Δ
nSlide19
Structure theorem for LTCs
We have shown: The constraint graph of an LTC is a small set expander.
Our main theorem shows that the constraint graph can be decomposed into
constant-
many (regular) expanders with few edges in between.Slide20
Main Theorem [
Dinur-K
]
Let C be an LTC with constraint graph G=(V,E). Then V can be partitioned into V
1
,…V
t
, such that
C =~ C
1 x C2 x … x Ct and such that each G(Vi) is an expander, with few edges between expanders. Moreover, each Ci is an LTC.t is at most 1/r.distanceExpansion depends on testability paramsSlide21
Summary and Questions
Expansion of a constraint graph implies distance.
Very high expansion of the constraint graph implies non-robustness.
The constraint graph of a robust code (LTC) is a small set expander for linear size sets. It can be decomposed into constant many (regular) expanders with few edges in between.
Q: What “amount” of expansion implies LTC?
Q: What structure of the constraint graph implies LTC?
Recent work (
Dinur-K
, and
Ben-Sasson-Viderman) shows that the constraint graph cannot be dense for LTCs with good rate.