Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer...
-
Upload
brook-pope -
Category
Documents
-
view
215 -
download
0
Transcript of Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer...
![Page 1: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/1.jpg)
Ryan O’Donnell
Carnegie Mellon University
a doctoral thesis, by
Per Austrin
KTH School of Computer Science
and Communication
opponent:
![Page 2: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/2.jpg)
Ryan O’Donnell
Carnegie Mellon University
a doctoral thesis, by
Per Austrin
KTH School of Computer Science
and Communication
opponent:
![Page 3: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/3.jpg)
Ryan O’Donnell
Carnegie Mellon University
opponent:
?
(Gödel Prize,
Royal Swedish
Acad. of Sci.)
(Gödel Prize x 2)
(Turing Award)
![Page 4: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/4.jpg)
Ryan O’Donnell
Carnegie Mellon University
opponent:
?
(Gödel Prize,
Royal Swedish
Acad. of Sci.)
(Gödel Prize x 2)
(Turing Award)
![Page 5: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/5.jpg)
Ryan O’Donnell
Carnegie Mellon University
opponent:
?
(Gödel Prize,
Royal Swedish
Acad. of Sci.)
(Gödel Prize x 2)
(Turing Award) Theoretical Computer Science:
![Page 6: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/6.jpg)
Theoretical Computer Science:
Which algorithmic problems
can be solved efficiently?
![Page 7: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/7.jpg)
Problem: 3Sat
Input:
Alg’s goal: an assignment satisfying as
many constraints as possible.
![Page 8: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/8.jpg)
“Efficient”
= “polynomial time”
= # steps always ≤ nC
![Page 9: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/9.jpg)
Input:
Obvious algorithm: ≈ 2n steps.
Question: Doable in nC steps?
![Page 10: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/10.jpg)
Answer: No.
Cook’s Theorem: 3Sat is “NP-hard”
NP-hard = Not doable in polynomial time
assuming “P ≠ NP”.
“P ≠ NP”: Everyone knows it’s true.
![Page 11: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/11.jpg)
Polynomial Time
Maximum Matching
Linear Programming
Primality
·····
1000’s of problems
NP-hard
3Sat
Traveling Salesperson
Chromatic Number
·····
1000’s of problems
an
y n
atu
ral p
rob
lem
s in
here
?
![Page 12: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/12.jpg)
Not known to be in P or NP-hard
1. Factoring
2. Graph Isomorphism
3. · · · · · ?Handbook on Algorithms and Theory of Computation [ALR99]:
“The vast majority of natural problems in NP have resolved themselves as being either in P or NP-complete. Unless you uncover a specific connection to one of [the above] intermediate problems, it is more likely offhand that your problem simply needs more work.”
NP-Completeness Column [Joh05]:
3. Precedence Constrained 3-Processor Scheduling
![Page 13: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/13.jpg)
Exact optimization
Approximation?
NP-hard
3Sat
Traveling Salesperson
Chromatic Number
·····
1000’s of problems
![Page 14: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/14.jpg)
Approximation?
95%-approximating 2Sat ?
90%-approximating 2CSP ?
15%-approximating 6CSP ?
Not known to be in P or to be NP-hard.Not known to be in P or to be NP-hard.
![Page 15: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/15.jpg)
Results from Austrin’s Thesis
95%-approximating 2Sat ? Hard.
90%-approximating 2CSP ? Hard.
15%-approximating 6CSP ? Hard.
![Page 16: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/16.jpg)
Results from Austrin’s Thesis
95%-approximating 2Sat ? Hard.*
90%-approximating 2CSP ? Hard.*
15%-approximating 6CSP ? Hard.*
* Not “NP-hard”, only “UG-hard”.
![Page 17: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/17.jpg)
Results from Austrin’s Thesis
95%-approximating 2Sat Hard.*
94.01656724%-approximating 2Sat Hard.*
Theorem [LLZ’02]:
94.01656724%-approximating 2Sat
can be done in polynomial time.αLLZ
αLLZ +
![Page 18: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/18.jpg)
Definition of αLLZ
= .9401656724…
![Page 19: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/19.jpg)
This is* the approximability threshold
of efficient algorithms for 2Sat!
![Page 20: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/20.jpg)
Remainder of the talk:
1. Definitions
2. Statements of main results
3. Remarks about proof techniques
![Page 21: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/21.jpg)
Remainder of the talk:
1. Definitions
2. Statements of main results
3. Remarks about proof techniques
![Page 22: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/22.jpg)
Max-CSP(P) Constraint Satisfaction Problem
![Page 23: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/23.jpg)
Max-CSP(P) Villkorssatisfieringsproblem
![Page 24: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/24.jpg)
Max-CSP(P) P : {0,1}k {acc, rej}
Input
“constraints”
![Page 25: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/25.jpg)
Max-CSP(P) P : {0,1}k {acc, rej}
Examples:
Max-kSat: P = “ORk”
Max-kLin: P = “XORk”
Max-kAND: P = “ANDk”
Max-kCSP: any mix of k-ary preds
![Page 26: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/26.jpg)
Max-CSP(P)
Many many many natural variants exist:
- constraints have different “weights”
- negated variables not allowed
- variables are {0, 1, 2, …, q-1}-valued
- have to use values {0, …, q-1}
“frugally”
![Page 27: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/27.jpg)
Approximation Algorithms
α-approximation algorithm:
On input I , guaranteed to output assignment
satisfying ≥ α · Opt(I ) constraints.
Goal: find poly-time such algorithms,
or, prove it’s NP-hard
![Page 28: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/28.jpg)
Approximation Algorithms
Trivial approximation for Max-CSP(P):
α-approximation, where
(Because choosing x1, …, xn randomly satisfies
α-fraction of all constraints in expectation.)
E.g.: (3/4)-approximation, for Max-2Sat.
![Page 29: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/29.jpg)
Approximation Algorithms
“Max-CSP(P) is approximation-resistant”:
= “Non-trivial approximation is NP-hard.”
E.g.: Max-3Sat is approximation-resistant.
[Håstad’97]
![Page 30: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/30.jpg)
Pairwise Independence
Let μ be a probability distribution on {0,1}k.
We say μ is pairwise independent if the
marginal on (Xi, Xj) is uniform on {0,1}2,
for all 1 ≤ i < j ≤ k, when (X1, …, Xk) ~ μ.
![Page 31: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/31.jpg)
UG-hard
A problem is said to be “UG-hard” if it is at least
as hard as the “Unique-Label-Cover Problem”.
UG Conjecture [Khot’02]:
“The Unique-Label-Cover Problem is NP-hard.”
Outstanding open problem in TCS,
b/c we don’t “know” the answer.
![Page 32: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/32.jpg)
Remainder of the talk:
1. Definitions
2. Statements of main results
3. Remarks about proof techniques
![Page 33: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/33.jpg)
Remainder of the talk:
1. Definitions
2. Statements of main results
3. Remarks about proof techniques
![Page 34: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/34.jpg)
Thesis Main Results
1. 2-CSP hardness
2. Approximation-resistant k-CSPs
3. Randomly supported pairwise independence
4. A technical result I’ll mention only briefly
![Page 35: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/35.jpg)
2-CSP Hardness
Let P : {0,1}2 {acc, rej}.
Let β(P) = min [somewhat complicated numerical program].
Then
β(P)-approximating Max-CSP(P) is UG-hard.
positive configuration family Θ
[Result 1]
![Page 36: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/36.jpg)
2-CSP Hardness
In particular:
• β(OR2) = αLLZ = .94016…
(matching the [LLZ’02] algorithm)
• β(AND2) ≤ .87434…,
(nearly matching the .87401…-approx.
algorithm for Max-CSP(AND2) [LLZ’02])
[Result 1]
![Page 37: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/37.jpg)
More on 2-ary Max-CSP(P)
Let β(P) = min [somewhat complicated numerical program].
Let α(P) = min [somewhat complicated numerical program].
Theorem: ∃ poly-time α(P)-approx alg.
Conjecture: α(P) = β(P) for all 2-ary P.
Then assuming the UG Conjecture,
β(P)-approximating Max-CSP(P) is hard.
positive configuration family Θ
all configuration families Θ
[Result 1]
![Page 38: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/38.jpg)
Presaged…
[Raghavendra’08]:
Let γ(P) = min [very complicated numerical program],
α(P) ≤ γ(P) ≤ β(P).
Theorem: ∃ poly-time γ(P)-approx alg.
and also (γ(P)+)-approximating is UG-hard.
Then assuming the UG Conjecture,
β(P)-approximating Max-CSP(P) is hard.
[Result 1]
![Page 39: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/39.jpg)
Approximation-resistant k-CSPs
Let P : {0,1}k {acc, rej}.
Suppose ∃ pairwise independent distribution
μ on {0,1}k such that supp(μ) ⊆ P-1(acc).
Then assuming the UG Conjecture,
Max-CSP(P) is approximation-resistant.
[Result 2,
with Mossel]
![Page 40: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/40.jpg)
Approximation-resistant k-CSPs
Q: How small a subset of {0,1}k can
support a pairwise independent distribution?
A: RoundUp4(k) points suffice
(assuming the Hadamard
Conjecture).
[Result 2]
![Page 41: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/41.jpg)
Approximation-resistant k-CSPs
( +)-UG-hardness for some 6-ary
CSP
( +)-UG-hardness for some 7-ary
CSP
( +)-UG-hardness for some k-ary
CSP
[Result 2]
Cor’s:
Previous best: , , NP-hardness [ST’00].
Best alg.: -approx. for Max-kCSP [CMM’07].
![Page 42: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/42.jpg)
Randomly supported pairwise independence
Q: Does a random subset of {0,1}k
of size S support a pairwise indep.
distr.?
Thm: Yes, whp, if S ≥ C · k2.
No, whp, if S ≤ c · k2.
[Result 3,
with Håstad]
![Page 43: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/43.jpg)
More generally…
Q: Does a random subset of {0, 1, …, q-1}k
of size S support a pairwise indep.
distr.?
Thm: Yes, whp, if S ≥ C(q) · k2.
No, whp, if S ≤ c(q) · k2.
[Result 3]
(& slightly weaker results for t-wise independence)
![Page 44: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/44.jpg)
Remainder of the talk:
1. Definitions
2. Statements of main results
3. Remarks about proof techniques
![Page 45: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/45.jpg)
Remainder of the talk:
1. Definitions
2. Statements of main results
3. Remarks about proof techniques
![Page 46: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/46.jpg)
Proof remarks for Result 3
Thm: C(q) k2 random pts in {0, 1, …, q-1}k
whp support a pairwise indep. distr.
Pf sketch: Need to show a certain random
convex body in ℝq2k2 contains origin whp.
Uses “hypercontractivity” to show that
quadratic polys of discrete rv’s are
fairly concentrated around expectation.
![Page 47: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/47.jpg)
Proofs for Hardness Results, 1 & 2
[Håstad’97] method for showing hardness:
PCP Technology Discrete Fourier
(“Label-Cover” is NP-hard) Analysis Wizardry+
![Page 48: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/48.jpg)
Proofs for Hardness Results, 1 & 2
Post-2002 method for showing hardness*:
PCP Technology Discrete Fourier
(“Label-Cover” is NP-hard) Analysis Wizardry+
UG Conjecture [Khot’02]
(“Unique-L-C is NP-hard”)
“Invariance Principle”
[MOO’05,Mos’08]
![Page 49: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/49.jpg)
Proofs for Hardness Results, 1 & 2
Post-2002 method had led to some new results:
• .87856… UG-hardness for “Max-Cut”
• UG-hardness of C-coloring 3-colorable
graphs (for all const C)
Based on “straightforward” use of Invariance Principle.
![Page 50: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/50.jpg)
Proofs for Hardness Results, 1 & 2
Key to Austrin’s new hardness results:
Heroically exploit the somewhat scary
Invariance Principle to its ultimate limits.
(Thesis Result 4: Preliminary work on Invariance
Principle generalization.)
![Page 51: Ryan O’Donnell Carnegie Mellon University a doctoral thesis, by Per Austrin KTH School of Computer Science and Communication opponent:](https://reader035.fdocuments.in/reader035/viewer/2022062713/56649ce05503460f949a9a7c/html5/thumbnails/51.jpg)
Thanks for your attention.
Time for questions?