linear programming and approximate constraint satisfaction

Transcription

1 linear programming and approximate constraint satisfaction Siu On Chan MSR New England James R. Lee University of Washington Prasad Raghavendra UC Berkeley David Steurer Cornell

2 results MAIN THEOREM: Any polynomial-sized linear program for problem MAX-CUT MAX-2SAT MAX-3SAT integrality gap 1/2 3/4 7/8 holds for LPs of size n c log n log log n MAIN TECHNIQUE: For approximating MAX-CSPs, polynomial-size LPs are exactly as powerful as those arising from O 1 rounds of the Sherali-Adams hierarchy.

3 a brief history of LP lower bounds Specific LP hierarchies (Lovász-Schrijver, Sherali-Adams) [Arora-Bollobás-Lovász 02] MAX-CUT has integrality gap 1/2 for Ω n rounds of LS [Schoenbeck-Trevisan-Tulsiani 07] ω 1 rounds of SA [Fernández de la Vega-Mathieu 07] n Ω 1 rounds of SA [Charikar-Makarychev-Makarychev 09] Extended formulations (EF) Every extended formulation for TSP has size 2 Ω n r-round relaxation has size no r [Yannakakis 88] every symmetric EF for TSP (and matching) has exponential size [Fiorini-Massar-Pokutta-Tiwary-de Wolf 12] EFs for approx. clique within n 1 2 ε require size 2 nε [Braun-Fiorini-Pokutta-Steurer 12] EFs for approx. clique within n 1 ε require size 2 nε [Braverman-Moitra 13]

4 a brief history of LP lower bounds Specific LP hierarchies (Lovász-Schrijver, Sherali-Adams) R N [Arora-Bollobás-Lovász 02] MAX-CUT has integrality gap 1/2 for Ω n rounds of LS [Schoenbeck-Trevisan-Tulsiani 07] ω 1 rounds of SA [Fernández de la Vega-Mathieu 07] n Ω 1 N n rounds of SA [Charikar-Makarychev-Makarychev 09] Extended formulations (EF) [Yannakakis 88] every symmetric EF for TSP (and matching) has exponential size Every extended formulation for TSP has size 2 Ω n r-round relaxation has size no r R n [Fiorini-Massar-Pokutta-Tiwary-de Wolf 12] EFs for approx. clique within n 1 2 ε require size 2 nε [Braun-Fiorini-Pokutta-Steurer 12] EFs for approx. clique within n 1 ε require size 2 nε [Braverman-Moitra 13]

5 what is a linear program for MAX-CUT? For a graph G = V, E and S V, write cut G S = so that opt G = max S V cut G S. E S, S E Standard relaxation: opt G = max x 1,1 n i j 1 x i x j Introduce variables {y ij } meant to represent 1 x i x j /2 2 max y ij subject to: i j {0 y ij 1} {y ij + y ik + y jk 2} {y ij y ik + y jk } {y ij + y jk + y kl + y lh + y hi 4}

6 what is a linear program for MAX-CUT? For a graph G = V, E and S V, write cut G S = so that opt G = max S V cut G S. Linearization: For every n, we have a natural number m and: - For every n-vertex graph G, a vector v G R m - For every cut S, a vector y S R m satisfying cut G S = v G, y S E S, S Relaxation: A polytope P R m such that y S P for every cut S The LP value is given by L G = max v G, x x P Size of the relaxation = # of inequalities needed to specify P E

7 approximation and integrality gaps An LP relaxation L is a (c, s)-approximation for MAX-CUT if for every graph G with opt G s, we have L G c. Linearization: For every n, we have a natural number m and: - For every n-vertex graph G, a vector v G R m - For every cut S, a vector y S R m satisfying cut G S = v G, y S Relaxation: A polytope P R m such that y S P for every cut S The LP value is given by L G = max v G, x x P Size of the relaxation = # of inequalities needed to specify P

8 approximation and integrality gaps An LP relaxation L is a (c, s)-approximation for MAX-CUT if for every graph G with opt G s, we have L G c. For the next theorem, view cut G as a function from 1,1 n to 0,1. THEOREM [Yannakakis via Farkas]: If there exists an LP relaxation of size R that is a (c, s)-approximation, then there are non-negative functions q 1, q 2,, q R : 1,1 n R + such that for every graph G with opt G s, there exists λ 1, λ 2,, λ R 0 satisfying c cut G = λ 1 q 1 + λ 2 q λ R q R

9 approximation and integrality gaps THEOREM [Yannakakis via Farkas]: If there exists an LP relaxation of size R that is a (c, s)-approximation, then there are non-negative functions q 1, q 2,, q R : 1,1 n R + such that for every graph G with opt G s, there exists λ 1, λ 2,, λ R 0 satisfying c cut G = λ 1 q 1 + λ 2 q λ R q R max v G, x b 1 A 1, x 0 b 2 A 2, x 0 b R A R, x 0 q i S = b i A i, y S Farkas Lemma says that every linear inequality valid for the polytope P can be derived from a nonnegative combination of the defining inequalities. Apply to the valid inequality c v G, x 0

10 lower bounds via separating hyperplanes THEOREM [Yannakakis via Farkas]: If there exists an LP relaxation of size R that is a (c, s)-approximation, then there are non-negative functions q 1, q 2,, q R : 1,1 n R + such that for every graph G with opt G s, there exists λ 1, λ 2,, λ R 0 satisfying c cut G = λ 1 q 1 + λ 2 q λ R q R Find a graph G and a hyperplane H such that: H, q i 0 for i = 1, 2, R, c cut G but H, c cut G < 0 H { 1,1} n R

11 the sherali-adams hierarchy THEOREM [Yannakakis via Farkas]: If there exists an LP relaxation of size R that is a (c, s)-approximation, then there are non-negative functions q 1, q 2,, q R : 1,1 n R + such that for every graph G with opt G s, there exists λ 1, λ 2,, λ R 0 satisfying c cut G = λ 1 q 1 + λ 2 q λ R q R A function q 1,1 n R is a k-junta if it only depends on k of its input coordinates. k rounds of Sherali-Adams corresponds to the case when all the q i s are k-junta s, i.e. c cut G cone(non-negative k-juntas)

12 junta reduction Let G 0 be a (c, s) gap instance for k rounds of Sherali-Adams. q 1, q 2,, q R : 1,1 n R + are n 0.2 -juntas c cut G0 G 0 H SA G 0 = m Works for R n 0.3k G = n m n

13 smoothing the q i s Normalize q 1, q 2,, q R : 1,1 n R + so that E q i = 1 Consider all the points at which q i x > R 2 for some i By Markov s inequality, total measure of such points is < 1 R Zero out the separating functional H on these points. Uses: H SA small

14 structure lemma LEMMA: Suppose q 1,1 n R + satisfies E q = 1 and q < R 2. Then there is an O k(log R) n 0.2 -junta q such that every degree-k Fourier coefficient of q q is at most n 0.1 Tells us nothing about the high-degree Fourier coefficients of q q. That s OK. The Sherali-Adams(k) functional H SA degree-k (as a multi-linear polynomial). 1,1 n R is

15 structure lemma LEMMA: Suppose q 1,1 n R + satisfies E q = 1 and q < R 2. Then there is an O k(log R) n 0.2 -junta q such that every degree-k Fourier coefficient of q q is at most n 0.1 q q Tells us nothing about the high-degree Fourier coefficients of q q. That s OK. The Sherali-Adams(k) functional H SA degree-k (as a multi-linear polynomial). 1,1 n R is H

16 recap q 1, q 2,, q R : 1,1 n R + c cut G (i) By zeroing H on a small set, can assume that E q i = 1 and q i < R 2 (ii) Every such q i can be approximated by an n 0.2 -junta q i so that q i q i has small degree-k Fourier coefficients. (iii) When randomly planting G 0, each q i becomes a k-junta on the support of G 0 G 0 H H { 1,1} n R (iv) The Sherali-Adams functional H SA is degree-k. Cannot see the high-degree discrepancy between q i and q i.

17 future directions For CSPs, does the connection between Sherali-Adams(k) and general LPs hold for k n ε? Can our method be extended beyond CSPs? (TSP, Vertex Cover, ) Can it be used to resolve the long-standing open problem: Do there exist polynomial-size extended formulations of the perfect matching polytope? Is there a similar connection between SDPs and the Lasserre hierarchy?