Approximability & Proof Complexity

1. Approximability& Proof Complexity Ryan O’Donnell Carnegie Mellon

2. Approximability & Proof Complexity of optimization problems Ryan O’Donnell Carnegie Mellon

3. Minimum Vertex-Cover Min-VC(G)= min {|S| : S ⊆ V, S touches all edges of E}

4. Minimum Vertex-Cover Min-VC(G)= min {|S| : S ⊆ V, S touches all edges of E}

5. “2-approximating” Min-VC • Choose any maximal matching M ⊆ E • MIN-VC(G) ≥ |M| • Let S = {all endpoints in M}.It’s a vertex-cover (why?) satisfying |S|= 2|M| ≤ 2MIN-VC(G)

6. “Factor 2-certifying” Min-VC • Choose any maximal matching M ⊆ E • Output “MIN-VC(G) ≥ |M|” A “factor α-certification” algorithm: • Output bound is always correct. • Bound is always within factor α of truth.

7. Linear programming (LP) relaxation k = minimize: ∑v∈V Xv subject to: 0 ≤ Xv≤ 1 for all v∈V Xu + Xv≥ 1 for all (u,v)∈E Output “MIN-VC(G) ≥ k”

8. Matching algorithm, LP algorithm are bothfactor 2-certification algorithms. Are they also 1.99-certification algorithms? No. Min-VC(Kn) = n−1 maximal |M| = n/2 LP value = n/2 Kn

9. Matching algorithm, LP algorithm are bothfactor 2-certification algorithms. Is there any poly-time 1.99-certification alg? Are they also 1.99-certification algorithms? No. Min-VC(Kn) = n−1 maximal |M| = n/2 LP value = n/2 Kn

10. Is there any poly-time 1.99-certification alg? We don’t know. Best we know is: ∃ 1.36-certification alg (unless P=NP) Is 1.99-certifying Min-VC NP-hard? [Dinur-Safra’02]

11. Approximability & Proof Complexity

12. Refutes statements encoded… Resolution by boolean disjunctions Cutting planes by integer inequalities Nullstellensatz/Polynomial Calculus[BIKPP’96,CEI’96] by polynomial equations Positivstellensatz/Sum-of-Squares (SOS) [Grigoriev-Vorobjov’99] by polynomial inequalities ZFC (“Frege”) in ordinary math language

13. Positivstellensatz[Krivine’64,Stengle’73,Schmüdgen’91,Putinar’93,Wörmann’98]Positivstellensatz[Krivine’64,Stengle’73,Schmüdgen’91,Putinar’93,Wörmann’98] assuming a mildtechnical condition is infeasible if and only if there exist “certifying polynomials” Q0, …, Qm, each a sum of squares, s.t. we have the identity −1 = Q0 + Q1P1 + Q2P2+ ••• +QmPm

14. “Min-VC(G) > k” for all v∈V ⇔ infeasible Xv2≥ XvXv2≤ Xv Xu+Xv≥ 1 for all (u,v)∈E ∑v Xv≤k ⇔ ∃ certifying SOS polynomials Q such that −1 = Q0 + Q1(k−∑ Xv) + ∑ Quv (Xu+Xv−1) + •••

15. Positivstellensatz / SOS proof system Suggested by Grigoriev and Vorobjov in 1999. The complexity of an SOS proof is the maximum degreeof any QiPi or Q0. SOSddenotes the proof systemrestricted to degree d. (No longer complete.)

16. Example proof Theorem: The following system is infeasible: {X2≤ 1, Y2≤ 1, Z2≤ 1, XY+YZ+ZX ≤ −2}. ZFC proof: Let f(X,Y,Z) = XY+YZ+ZX. Suppose X2≤ 1; i.e., X ∈ [−1,1]. Since f is linear in X, it’s maximized if X = ±1. Similarly for Y and Z. Suffices to show f(±1,±1,±1) > −2. If all three inputs same, f is 3. If not all three inputs same, f is −1.

17. Example proof Theorem: The following system is infeasible: {X2≤ 1, Y2≤ 1, Z2≤ 1, XY+YZ+ZX ≤ −2}. SOSd=4 proof: +

18. Exercise Show SOSd=4 certifies MIN-VC(Kn) ≥ n−1 (i.e., refutes MIN-VC(Kn) ≤ n−2).

19. SOSd is ‘automatizable’! Theorem:[Lasserre’00,Parrilo’00, cf. N.Shor’87] If a polynomial inequality system can be refuted in theSOSdproof system, the certifying Qi’s can be found innO(d) time (using semidefinite programming).

20. The strongest(?) automatizableproof system that we know SOSd is stronger than: Width-d Resolution Degree-d NullstellensatzBasic LP relaxationsBasic SDP relaxationsd/2 rounds of Lovász-Schrijver LP/SDP hierarchyd/2 rounds of Sherali-Adams LP/SDP hierarchy (Doesn’t seem to be stronger than degree-d Polynomial Calculus.)

21. A very powerful poly-time algorithmfor Vertex-Cover certification: Output the largest k ∈ [n] such thatSOSd=1000 certifies “Min-VC(G) > k”. Could this be a 1.99-certification algorithm? I.e., is it true that whenever MIN-VC(G) = β,∃ degree-1000 Qi’s certifying Min-VC(G) ≥β/1.99?

22. Partial history of upperand lower bounds for SOSd

23. SOSd upper bounds, 2001-2011 Nothing that we didn’t alreadyknow by other means. E.g., SDP is a .878-certification alg for Max-Cut ∵ SDP ≤ SOSd=4 ∴ SOSd=4 also .878-certifies Max-Cut

24. SOSd lower bounds, 1999-2009

25. Tseitin Tautologies / 3Lin(mod 2) Consider a random system of O(n)3-variable equations modulo 2. [Grigoriev’99]:(indep. [Schoenbeck’08]) With very high probability… • No assignment sats > 51% of equations • Unless d = Ω(n), SOSd cannot refute “the system is totally satisfiable”.

26. ‘Knapsack’ “If n is odd and X1, …, Xn satisfy Xi2 = 1, then X1 + ••• + Xn cannot be 0.” (Essentially equivalent: “MAX-CUT(Kn) ≥ ”) [Grigoriev’01]: See also [Laurent’02],[Cheung’05] Not provable in SOSdunless d ≥n+1. Open Problem: Give a pleasant proof that d needs to be at least, say, 6.

27. [Tulsiani’09] Rule of Thumb (A corollary of the 3Lin(mod 2) lower bound.) (Not rigorously proven, but seems true in all cases.) For any factor-α certification problemwhich we know is NP-hard, there exists instances which requiredegree-nΩ(1) SOS proofs.

28. A very powerful poly-time algorithmfor Vertex-Cover certification: Output the largest k ∈ [n] such thatSOSd=1000 certifies “Min-VC(G) > k”. Could this be a 1.99-certification algorithm?

29. Integrality Gaps [GK’95] SDP does not 2-certify Min-VC: Frankl-Rödl graphs [FS’00] SDP does not .879-certify Max-Cut: Noisy-sphere graphs [KV’05] SDP+∆ does not .879-certify Max-Cutor solve Unique-Games: KV noisy-hypercube graphs DKSV noisy-hypercube graphs [DKSV’06] SDP+∆ does not O(1)-certifySparsest-Cut (Balanced-Separator): [KS’09,RS’09] Sherali-Adams+, degree-O(1),does not .879-certify Max-Cutor solve Unique-Games: KV noisy-hypercube graphs [BCGM’11] Sherali-Adams+, degree-6, (and probably degree-O(1)) does not 2-certify Min-VC Frankl-Rödl graphs

30. These tricky instancesaren’t so hard for SOS!

31. [BBHKSZ’12]: SOSd=4 solves the KV Unique-Games instances! [OZ’13]: SOSd=4 solves the DKSV Balanced-Separator instances. [OZ’13]: SOSd=O(1) .95-certifies the KV Max-Cut instances.

32. [BBHKSZ’12]: SOSd=4 solves the KV Unique-Games instances! [OZ’13]: SOSd=4 solves the DKSV Balanced-Separator instances. [DMN’13]: SOSd=O(1) solves the KV Max-Cut instances. • [KV’05]:used ZFC to show “MAX-CUT(KV) ≈ 85%” • [KS’09,RS’09]:SA+d=O(1)only certify “MAX-CUT(KV) ≥ 75%” • [DMN’13] SOSdcertifies “MAX-CUT(KV) ≥ 85% − od(1)”

33. [BBHKSZ’12]: SOSd=4 solves the KV Unique-Games instances! [OZ’13]: SOSd=4 solves the DKSV Balanced-Separator instances. [DMN’13]: SOSd=O(1) solves the KV Max-Cut instances. [KOTZ’13]: SOSd=O(1) solves “most of” the Frankl-Rödl Min-VC instances.

34. The whole result is just thatone particular algorithm doeswell on one particular instance? I have 3 responses. 

35. Response 1: an old joke Q: Why did the complexity theorist work on algorithms? A: To get lower bounds on his lower bounds. We basically no longer know any “hard instances”.

36. Response 2: Evidence for algorithmic optimism? [BBHKSZ+’13] points out that as far as we know, SOSd=4 solves the Unique-Games problem (i.e., refutes the UGC). Perhaps SOSd is the killer algorithmfor combinatorial optimization.

37. Response 3: New proofs Proving known theorems in restricted proofsystems can lead to new insights and proofs. [Razborov’93]: New Switching Lemma proof [BBHKSZ’12, Hypercontractivity insights OZ’13,KOTZ’13]: [BHM’12,KOTZ’13]: New Frankl-Rödl Thm. proof [MN’13,DMN’13]: New Maj. is Stablest proof

38. Let’s take stock • Approximation Algs ≤ Proof Complexity:“Is there an efficient algorithm for 100-coloring a 3-colorable graph?”

39. Let’s take stock • Approximation Algs ≤ Proof Complexity: • “Given a graph that is not 100-colorable, how hard is it to prove that it’s not 3-colorable?” • SOSd is a quirky yet strong proof system,automatizable in time nO(d). • SOSd=O(1) solves all the trickiest instanceswe know of Unique-Games, Max-Cut.

40. Three closing thoughtsregarding proof complexity • Frankl-Rödl graphs and SOS lower bounds • The Dynamic SOS proof system • My favorite algorithm for Unique-Games

41. Frankl-Rödl graphs FRm(γ): V = {0,1}m E = {(x,y) : Δ(x,y) ≥ (1−γ)m} [Frankl-Rödl’87]: MIN-VC(FRm(γ)) ≥(1−o(1))2m [KOTZ’13]: SOSd=O(1/γ)can prove this. But perhaps SOSd=O(1)cannot handle

42. A simpler open problem Theorem: (a corollary of [Harper’66]’s Vertex-Isoperimetric Inequality) Let A, B ⊆ {0,1}m satisfy dist(A,B) ≥ Then |A|, |B| aren’t both large: Conjecture: SOSd=4 cannotprove this.

43. Dynamic SOS • Lines of the proof are of form P(X1, …, Xn) ≥ 0. • From P ≥ 0 and Q ≥ 0 can derive P + Q ≥ 0 and P • Q ≥ 0. • Can always derive R2≥ 0. • To refute a system {P1≥ 0, …, Pm≥ 0},derive −1 ≥ 0. Complexity: max degree of any line

44. Dynamic SOS Facts [Grigoriev-Hirsch-Pasechnik’01]: • Dynamic SOSd=3 refutes Knapsack • Dynamic SOSd=5 refutes any unsatisfiable 3XOR instance Open problem 1 [GHP’01]: Suggest an explicit unsatisfiable boolean formula which SOSd=O(1) might not refute. Open problem 2: Give negative evidence re automatizability.

45. Unique-Games [Khot’02] conjectured that for the “UG” CSP, it’s NP-hard to distinguish ϵ-satisfiable instances from (1−ϵ)-satisfiable instances. [BBHKSZ’12]: Perhaps SOSd=4 can actually do it.⇒ UGC is false (assuming NP ≠ P) Perhaps SOSd=log(n) can do it.⇒ UGC is false (assuming NP ⊈ TIME[nlog n]) Why be so concerned about automatizability?

46. Unique-Games [Khot’02] conjectured that for the “UG” CSP, it’s NP-hard to distinguish ϵ-satisfiable instances from (1−ϵ)-satisfiable instances. My favorite UG algorithm: Given an ϵ-satisfiable instance, nondeterministically guess a poly-lengthZFC proof that instance is ≤ (1−ϵ)-satisfiable. If this algorithm works, UGC is false.(assuming NP ≠ coNP)

47. ありがとうございました！Thank you!