Linear vs. semidefinite extended formulations

1. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds Linear vs. Semideﬁnite Extended Formulations: Exponential Separation and Strong Lower Bounds Samuel Fiorini Serge Massar Sebastian Pokutta ULB/Math ULB/Phys Erlangen U. Hans Raj Tiwary Ronald de Wolf ULB/Math CWI Sebastian Pokutta Linear vs. Semideﬁnite Extended Formulations 1

2. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds History In 86-87, Swart claimed he could prove P = NP How? By giving a poly-size linear program (LP) for the TSP Theorem (Yannakakis’88/91) Every symmetric LP for the TSP has size 2Ω(n) Swart’s LP was symmetric and of size poly(n) =⇒ it was wrong Sebastian Pokutta Linear vs. Semideﬁnite Extended Formulations 2

7. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds History From Yannakakis’91: Sebastian Pokutta Linear vs. Semideﬁnite Extended Formulations 3

8. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds History From Yannakakis’11: Sebastian Pokutta Linear vs. Semideﬁnite Extended Formulations 3

9. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds Recent previous results Deﬁnition P, Q polytopes. Q is an EF of P if ∃ linear π with π(Q) = P. Size of Q := #facets of Q. Q π P • Kaibel, Pashkovich & Theis’10: some polytopes have no poly-size symmetric EF but poly-size non-symmetric EFs. • Rothvoß’11: there are 0/1-polytopes in Rd such that every EF has size 2(1/2−o(1))d . Sebastian Pokutta Linear vs. Semideﬁnite Extended Formulations 4

12. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds Our results We show that Every LP for the TSP has super-polynomial size. via the following sequence of reductions: 1/4 • every EF of the TSP polytope has size 2Ω(n ) ⇑ 1/2 • ∃ (Gn )n s.t. every EF of STAB(Gn ) has size 2Ω(n ) ⇑ • every EF of the cut polytope has size 2Ω(n) This is the 1st outcome of a new “SDP←→quantum” connection Remark. Generalizes “linear EF ←→ classical CC” connection (Faenza, Fiorini, Grappe, Tiwary’11) Today: We focus on the TSP result! Sebastian Pokutta Linear vs. Semideﬁnite Extended Formulations 5

19. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds Extension complexity and slack matrices Consider a polytope P (with dim(P) 1) Definition (Extension complexity) Extension complexity of P, xc(P) := minimum size of EF of P Let A ∈ Rm×d , b ∈ Rm , V = {v1 , . . . , vn } ⊆ Rd s.t. P = {x ∈ Rd | Ax b} = conv(V ) Definition (Slack matrix) Slack matrix S ∈ Rm×n of P w.r.t. Ax + b and V : Sij := bi − Ai vj Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 6

23. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds Nonnegative factorizations and the factorization theorem Deﬁnition (Nonnegative factorization and rank) A rank-r nonnegative factorization of S ∈ Rm×n is S = TU where T ∈ Rm×r + and U ∈ Rr ×n + Nonnegative rank of S: rk+ (S) := min{r | ∃ rank-r nonnegative factorization of S} Theorem (Yannakakis’91) For every slack matrix S of P: xc(P) = rk+ (S) Sebastian Pokutta Linear vs. Semideﬁnite Extended Formulations 7

26. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds Bounding the extension complexity: the Rectangle Covering Bound S = TU nonnegative factorization = T k Uk sum of nonnegative rank-1 matrices k∈[r ] =⇒ supp(S) = supp(T k Uk ) k∈[r ] = supp(T k ) × supp(Uk ) union of rectangles (as sets) k∈[r ] Deﬁnition (Rectangle covering number rc(S)) rc(S) := min{r | ∃ rectangle covering of size r }. Sebastian Pokutta Linear vs. Semideﬁnite Extended Formulations 8

29. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds Bounding the extension complexity: the Rectangle Covering Bound 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 Sebastian Pokutta Linear vs. Semideﬁnite Extended Formulations 9

30. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds Bounding the extension complexity: the Rectangle Covering Bound 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 Sebastian Pokutta Linear vs. Semideﬁnite Extended Formulations 9

31. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds Bounding the extension complexity: the Rectangle Covering Bound Observation. rk+ (S) rc(S) (Yannakakis’91) Observation. For every polytope P and slack matrix S of P: xc(P) = rk+ (S) rc(S) = rc( suppmat(S) ) nonincidence matrix (Fiorini, Kaibel, Pashkovich & Theis’11) Remark. This is related to nondeterministic CC (= log rc(M) ) Sebastian Pokutta Linear vs. Semideﬁnite Extended Formulations 10

34. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds Lower bound for the correlation polytope / cut polytope Let M = M(n) be the 2n × 2n matrix with Mab := (1 − aT b)2 for a, b ∈ {0, 1}n Remark. • Not a slack matrix, but can be embedded in a slack matrix • Support matrix appears in de Wolf’03 for separating classical vs. quantum nondeterministic complexity Sebastian Pokutta Linear vs. Semideﬁnite Extended Formulations 11

35. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds Lower bound for the correlation polytope / cut polytope Let M = M(n) be the 2n × 2n matrix with Mab := (1 − aT b)2 for a, b ∈ {0, 1}n Remark. • Not a slack matrix, but can be embedded in a slack matrix • Support matrix appears in de Wolf’03 for separating classical vs. quantum nondeterministic complexity Sebastian Pokutta Linear vs. Semideﬁnite Extended Formulations 11

36. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds Lower bound for the correlation polytope / cut polytope Observation. For b ∈ {0, 1}n : 1 − 2diag(a) − aaT , bb T = 1 − 2 diag(a), bb T + aaT , bb T = 1 − 2 diag(a), diag(b) + aaT , bb T = 1 − 2 aT b + (aT b)2 = (1 − aT b)2 = Mab Correlation polytope: COR(n) := conv{bb T ∈ Rn×n | b ∈ {0, 1}n } Lemma (Key Lemma) For every a ∈ {0, 1}n , the inequality ( ) 2diag(a) − aaT , x 1 is valid for COR(n). The slack of vertex bb T w.r.t. ( ) is Mab . Sebastian Pokutta Linear vs. Semideﬁnite Extended Formulations 12

37. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds Lower bound for the correlation polytope / cut polytope Observation. For b ∈ {0, 1}n : 1 − 2diag(a) − aaT , bb T = 1 − 2 diag(a), bb T + aaT , bb T = 1 − 2 diag(a), diag(b) + aaT , bb T = 1 − 2 aT b + (aT b)2 = (1 − aT b)2 = Mab Correlation polytope: COR(n) := conv{bb T ∈ Rn×n | b ∈ {0, 1}n } Lemma (Key Lemma) For every a ∈ {0, 1}n , the inequality ( ) 2diag(a) − aaT , x 1 is valid for COR(n). The slack of vertex bb T w.r.t. ( ) is Mab . Sebastian Pokutta Linear vs. Semideﬁnite Extended Formulations 12

38. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds Lower bound for the correlation polytope / cut polytope Consider complete linear description for COR(n) starting with 2diag(a) − aaT , x 1 ∀a ∈ {0, 1}n and corresponding slack matrix S bb T 2diag(a) − aaT , x 1 M S xc(COR(n)) = rk+ (S) rc(S) rc(M) = 2Ω(n) (de Wolf’03 using Razborov’92) Sebastian Pokutta Linear vs. Semideﬁnite Extended Formulations 13

41. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds Lower bound for the correlation polytope / cut polytope As CUT(n) ∼ COR(n − 1) (aﬃne) we obtain: = Theorem (Cut polytope) xc(CUT(n)) = xc(COR(n − 1)) = 2Ω(n) . This is our problem-0 and we do have a reduction mechanism: Lemma (Monotonicity) • Q is an EF of P =⇒ xc(Q) xc(P) • P contains F as a face =⇒ xc(P) xc(F ) Sebastian Pokutta Linear vs. Semideﬁnite Extended Formulations 14

44. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds Lower bound for the stable set polytope Stable set polytope. ∀ k ∃ Hk with O(k 2 ) vertices s.t. STAB(Hk ) has a face F = F (k) that is an EF of COR(k). u u e e e e xc(STAB(Hk )) xc(F (k)) xc(COR(k)) = 2Ω(k) v v Theorem (Stable set polytope) For all n there exists an n-vertex graph Gn s.t. 1/2 ) xc(STAB(Gn )) = 2Ω(n Sebastian Pokutta Linear vs. Semideﬁnite Extended Formulations 15

47. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds Lower bound for the TSP polytope TSP polytope. ∀ k-vertex G , one can ﬁnd face F = F (k) of TSP(n) with n = O(k 2 ) such that F (k) is an EF of STAB(G ). (Yannakakis’91). Theorem (TSP polytope) 1/4 ) xc(TSP(n)) = 2Ω(n Sebastian Pokutta Linear vs. Semideﬁnite Extended Formulations 16

48. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds Thank you! Sebastian Pokutta Linear vs. Semideﬁnite Extended Formulations 17

Linear vs. semidefinite extended formulations

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Linear vs. semidefinite extended formulations