1. Conditional Inapproximability and Limited Independence
a doctoral thesis, by
Per Austrin
KTH School of Computer Science
and Communication
opponent:
Ryan O’Donnell
Carnegie Mellon University

2. Conditional Inapproximability and Limited Independence
a doctoral thesis, by
Per Austrin
KTH School of Computer Science
and Communication
opponent:
Ryan O’Donnell
Carnegie Mellon University

3. Carnegie Mellon University
(Turing Award)
(Gödel Prize x 2)
(Gödel Prize, Royal Swedish Acad. of Sci.) ?
opponent:
Ryan O’Donnell
Carnegie Mellon University

4. Carnegie Mellon University
(Turing Award)
(Gödel Prize x 2)
(Gödel Prize, Royal Swedish Acad. of Sci.) ?
opponent:
Ryan O’Donnell
Carnegie Mellon University

5. Carnegie Mellon University
Theoretical Computer Science:
(Turing Award)
(Gödel Prize x 2)
(Gödel Prize, Royal Swedish Acad. of Sci.) ?
opponent:
Ryan O’Donnell
Carnegie Mellon University

6. Theoretical Computer Science:
Which algorithmic problems
can be solved efficiently?

7. Problem: 3Sat
Input:
Alg’s goal: an assignment satisfying as many constraints as possible.

8. “Efficient”
= “polynomial time”
= # steps always ≤ nC

9. Question: Doable in nC steps?
Input:
Obvious algorithm: ≈ 2n steps.
Question: Doable in nC steps?

10. 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.

11. Traveling Salesperson
Polynomial Time
Maximum Matching
Linear Programming
Primality
·····
1000’s of problems
NP-hard
3Sat
Traveling Salesperson
Chromatic Number
·····
1000’s of problems
any natural problems in here?

12. 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 Processor Scheduling

13. Traveling Salesperson
Exact optimization
Approximation?
NP-hard
3Sat
Traveling Salesperson
Chromatic Number
·····
1000’s of problems

14. Not known to be in P or to be NP-hard.
Approximation?
95%-approximating 2Sat ?
90%-approximating 2CSP ?
15%-approximating 6CSP ?
Not known to be in P or to be NP-hard.

15. Results from Austrin’s Thesis
95%-approximating 2Sat ? Hard.
90%-approximating 2CSP ? Hard.
15%-approximating 6CSP ? Hard.

16. * Not “NP-hard”, only “UG-hard”.
Results from Austrin’s Thesis
95%-approximating 2Sat ? Hard.*
90%-approximating 2CSP ? Hard.*
15%-approximating 6CSP ? Hard.*
* Not “NP-hard”, only “UG-hard”.

17. Results from Austrin’s Thesis
95%-approximating 2Sat Hard.*
%-approximating 2Sat Hard.*
Theorem [LLZ’02]:
%-approximating 2Sat can be done in polynomial time.
αLLZ + 
αLLZ

18. Definition of αLLZ
= …

19. This is* the approximability threshold of efficient algorithms for 2Sat!

20. Remainder of the talk: 1. Definitions 2. Statements of main results
3. Remarks about proof techniques

21. Remainder of the talk: 1. Definitions 2. Statements of main results
3. Remarks about proof techniques

22. Max-CSP(P) Constraint Satisfaction Problem
villkorssatisfieringsproblem,
P is a predicate on k binary inputs

23. Max-CSP(P) Villkorssatisfieringsproblem villkorssatisfieringsproblem,
P is a predicate on k binary inputs

24. Max-CSP(P) P : {0,1}k  {acc, rej}
Input “constraints”
villkorssatisfieringsproblem,
P is a predicate on k binary inputs

25. 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

26. 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”
ax-Cut, Vertex-Cover, Graph-Coloring, Sparsest-Cut, Max-Clique, …

27. Approximation Algorithms
On input I , guaranteed to output assignment satisfying ≥ α · Opt(I ) constraints.
Goal: find poly-time such algorithms,
or, prove it’s NP-hard

28. 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.
Do example: Max-2Sat

29. Approximation Algorithms
“Max-CSP(P) is approximation-resistant”:
= “Non-trivial approximation is NP-hard.”
E.g.: Max-3Sat is approximation-resistant.
[Håstad’97]

30. 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.
t-wise independence, non-uniform marginals, etc.

31. 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.
Defining the “Unique-Label-Cover Problem” now would kind of kill the rhetorical flow.

32. Remainder of the talk: 1. Definitions 2. Statements of main results
3. Remarks about proof techniques

33. Remainder of the talk: 1. Definitions 2. Statements of main results
3. Remarks about proof techniques

34. 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

35. 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.
[Result 1]
positive configuration family Θ

36. 2-CSP Hardness In particular:
β(OR2) = αLLZ = … (matching the [LLZ’02] algorithm)
β(AND2) ≤ …, (nearly matching the …-approx. algorithm for Max-CSP(AND2) [LLZ’02])
[Result 1]

37. 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.
[Result 1]
positive configuration family Θ
all configuration families Θ

38. 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]

39. 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]

40. 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]

41. 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].

42. Randomly supported pairwise independence
[Result 3, with Håstad]
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.

43. Thm: Yes, whp, if S ≥ C(q) · k2. No, whp, if S ≤ c(q) · k2.
[Result 3]
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.
(& slightly weaker results for t-wise independence)

44. Remainder of the talk: 1. Definitions 2. Statements of main results
3. Remarks about proof techniques

45. Remainder of the talk: 1. Definitions 2. Statements of main results
3. Remarks about proof techniques

46. 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.

47. Proofs for Hardness Results, 1 & 2
[Håstad’97] method for showing hardness:
PCP Technology Discrete Fourier
(“Label-Cover” is NP-hard) Analysis Wizardry +

48. 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]

49. Proofs for Hardness Results, 1 & 2
Post-2002 method had led to some new results:
• … UG-hardness for “Max-Cut” • UG-hardness of C-coloring 3-colorable graphs (for all const C)
Based on “straightforward” use of Invariance Principle.

50. 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.)

51. Thanks for your attention.
Time for questions?