Computing the Independence Polynomial: from the Tree Threshold Down to the Roots

Transcription

1 1 / 16 Computing the Independence Polynomial: from the Tree Threshold Down to the Roots Nick Harvey 1 Piyush Srivastava 2 Jan Vondrák 3 1 UBC 2 Tata Institute 3 Stanford SODA 2018

2 The Lovász Local Lemma 2 / 16 Theorem (Symmetric form, Erdős-Lovász 75) If E 1, E 2,... are events on a probability space Ω such that Each event is independent of all but d other events 1 The probability of each event is at most e(d+1) then [ ] Pr E i > 0 (but typically exponentially small) i

3 Example: 2-colorability of hypergraphs 3 / 16 Given: a system of sets, each of size k, and each intersecting at most d := 1 2e 2k sets. Goal: color the elements red/blue so that every set contains both colors.

4 Example: 2-colorability of hypergraphs 3 / 16 Given: a system of sets, each of size k, and each intersecting at most d := 1 2e 2k sets. Goal: color the elements red/blue so that every set contains both colors. Randomized coloring: Color elements independently red/blue.

5 Example: 2-colorability of hypergraphs 3 / 16 Given: a system of sets, each of size k, and each intersecting at most d := 1 2e 2k sets. Goal: color the elements red/blue so that every set contains both colors. Randomized coloring: Color elements independently red/blue. LLL analysis: For set A i, event E i occurs if A i is all-red or all-blue. Each event E i is independent of all E j where A i A j =. Parameter d chosen s.t. LLL Pr[no set is monochromatic] > 0

6 4 / 16 History of the Algorithmic LLL Many attempts to make the LLL algorithmic: Some applications algorithmic, with weaker parameters [Beck 91], [Alon 91], [Molloy-Reed 98], [Czumaj-Scheideler 00], [Srinivasan 08]...

7 4 / 16 History of the Algorithmic LLL Many attempts to make the LLL algorithmic: Some applications algorithmic, with weaker parameters [Beck 91], [Alon 91], [Molloy-Reed 98], [Czumaj-Scheideler 00], [Srinivasan 08]... Moser s Algorithm [2008]: As long as there is a monochromatic set, pick an arbitrary one and re-color it randomly.

8 4 / 16 History of the Algorithmic LLL Many attempts to make the LLL algorithmic: Some applications algorithmic, with weaker parameters [Beck 91], [Alon 91], [Molloy-Reed 98], [Czumaj-Scheideler 00], [Srinivasan 08]... Moser s Algorithm [2008]: As long as there is a monochromatic set, pick an arbitrary one and re-color it randomly.

9 4 / 16 History of the Algorithmic LLL Many attempts to make the LLL algorithmic: Some applications algorithmic, with weaker parameters [Beck 91], [Alon 91], [Molloy-Reed 98], [Czumaj-Scheideler 00], [Srinivasan 08]... Moser s Algorithm [2008]: As long as there is a monochromatic set, pick an arbitrary one and re-color it randomly.

10 4 / 16 History of the Algorithmic LLL Many attempts to make the LLL algorithmic: Some applications algorithmic, with weaker parameters [Beck 91], [Alon 91], [Molloy-Reed 98], [Czumaj-Scheideler 00], [Srinivasan 08]... Moser s Algorithm [2008]: As long as there is a monochromatic set, pick an arbitrary one and re-color it randomly. Theorem: [Moser-Tardos 08] If each set intersects at most 2 k /2e others, this algorithm terminates in expected polynomial time.

11 Algorithm dynamics 5 / 16 Previous LLL algorithms: resampling infeasible solutions Space {0, 1} m of colorings Feasible region: good colorings Algorithm randomly generates integral points until feasible.

12 Algorithm dynamics 5 / 16 Previous LLL algorithms: resampling infeasible solutions Space {0, 1} m of colorings Feasible region: good colorings Algorithm randomly generates integral points until feasible.

13 Algorithm dynamics 5 / 16 Previous LLL algorithms: resampling infeasible solutions Space {0, 1} m of colorings Feasible region: good colorings Algorithm randomly generates integral points until feasible.

14 Algorithm dynamics 5 / 16 Previous LLL algorithms: resampling infeasible solutions Space {0, 1} m of colorings Feasible region: good colorings Algorithm randomly generates integral points until feasible.

15 Algorithm dynamics 5 / 16 Previous LLL algorithms: resampling infeasible solutions Space {0, 1} m of colorings Feasible region: good colorings Algorithm randomly generates integral points until feasible.

16 Algorithm dynamics 5 / 16 Our LLL algorithm: rounding feasible solution Space {0, 1} m of colorings Feasible region: good colorings Starts with feasible, fractional coloring

17 Algorithm dynamics 5 / 16 Our LLL algorithm: rounding feasible solution Space {0, 1} m of colorings Feasible region: good colorings Starts with feasible, fractional coloring Algorithm rounds each coordinate to integrality, maintaining feasibility.

18 Algorithm dynamics 5 / 16 Our LLL algorithm: rounding feasible solution Space {0, 1} m of colorings Feasible region: good colorings Starts with feasible, fractional coloring Algorithm rounds each coordinate to integrality, maintaining feasibility.

19 Algorithm dynamics 5 / 16 Our LLL algorithm: rounding feasible solution Space {0, 1} m of colorings Feasible region: good colorings Starts with feasible, fractional coloring Algorithm rounds each coordinate to integrality, maintaining feasibility.

20 Algorithm dynamics 5 / 16 Our LLL algorithm: rounding feasible solution Space {0, 1} m of colorings Feasible region: good colorings Starts with feasible, fractional coloring Algorithm rounds each coordinate to integrality, maintaining feasibility. What is this feasible region?

21 The Feasible Region 6 / 16 Dependency Graph G: Vertex v in G bad event E v set A v. Edges in G dependent events sets that intersect. Maximum degree of G d (amount of dependence).

22 The Feasible Region 6 / 16 Dependency Graph G: Vertex v in G bad event E v set A v. Edges in G dependent events sets that intersect. Maximum degree of G d (amount of dependence). Observation: The sets containing a particular element form a clique in G.

23 The Feasible Region 6 / 16 Dependency Graph G: Vertex v in G bad event E v set A v. Edges in G dependent events sets that intersect. Maximum degree of G d (amount of dependence). Observation: The sets containing a particular element form a clique in G. Independence Polynomial: q(p) = ( p v ), indep. I where p v [0, 1] are per-vertex parameters. v I

24 The Feasible Region 6 / 16 Dependency Graph G: Vertex v in G bad event E v set A v. Edges in G dependent events sets that intersect. Maximum degree of G d (amount of dependence). Observation: The sets containing a particular element form a clique in G. Independence Polynomial: q(p) = ( p v ), indep. I where p v [0, 1] are per-vertex parameters. The Feasible Region for the LLL: The Shearer region { } S = p [0, 1] V : q(p ) > 0 0 p p. v I

25 The Feasible Region 7 / 16 Independence Polynomial: q(p) = Shearer region: S = ( p v ), p v [0, 1] V indep. I v I { } p [0, 1] V : q(p ) > 0 0 p p Theorem (Shearer 85) Let E 1, E 2,... be events with p i = Pr[E i ]. Let G be a graph s.t. each event is independent of its non-neighbors in G p S. Then Pr[ i E i] q(p) > 0. False for all p S.

26 A Simple Case: The Univariate Feasible Region 8 / 16 Independence Polynomial: q(p) = ( p v ), p v [0, 1] V indep. I v I { } Shearer region: S = p [0, 1] V : q(p ) > 0 0 p p Closest root to origin for G: Let λ G = min { z : q(z1) = 0 }. 0 no roots λ G R

27 A Simple Case: The Univariate Feasible Region 8 / 16 Independence Polynomial: q(p) = ( p v ), p v [0, 1] V indep. I v I { } Shearer region: S = p [0, 1] V : q(p ) > 0 0 p p Closest root to origin for G: Let λ G = min { z : q(z1) = 0 }. Shearer s lemma: If all p i λ G then p S, so Pr[ i E i] > 0. 0 Pr[ i E i] > 0 Pr[ i E i] = 0 (in some Ω) no roots λ G R

28 A Simple Case: The Univariate Feasible Region Independence Polynomial: q(p) = ( p v ), p v [0, 1] V indep. I v I { } Shearer region: S = p [0, 1] V : q(p ) > 0 0 p p Closest root to origin for G: Let λ G = min { z : q(z1) = 0 }. Shearer s lemma: If all p i λ G then p S, so Pr[ i E i] > 0. 0 Pr[ i E i] > 0 Pr[ i E i] = 0 (in some Ω) no roots (in any max-degree d graph) λ d 1 ed λ G R Closest root to origin for degree d graphs: Let λ d := inf G λ G over all graphs of maximum degree d. Then λ d = (d 1)d 1 d d 1 ed. 8 / 16

29 A rounding approach to LLL (for hypergraph coloring) 1: procedure LLLBYROUNDING 2: Initial fractional coloring: each element 1 2 red and 1 2 blue 3: for i 1,..., m do 4: Color element i red or blue, whichever preserves feasibility in S 5: return integer coloring Required to prove: For existential LLL: coloring element either red or blue preserves feasibility in S For algorithmic LLL: can efficiently decide feasibility for S 9 / 16

30 A rounding approach to LLL (for hypergraph coloring) 1: procedure LLLBYROUNDING 2: Initial fractional coloring: each element 1 2 red and 1 2 blue 3: for i 1,..., m do 4: Color element i red or blue, whichever preserves feasibility in S 5: return integer coloring Required to prove: For existential LLL: coloring element either red or blue preserves feasibility in S For algorithmic LLL: can efficiently decide feasibility for S 9 / 16

31 Preserving feasibility: a proof of existential LLL 10 / 16 Recall that q(p) = ( p v ) indep. I v I { } S = p [0, 1] V : q(p ) > 0 0 p p

32 Preserving feasibility: a proof of existential LLL 10 / 16 Recall that q(p) = indep. I ( p v ) v I { } S = p [0, 1] V : q(p ) > 0 0 p p { } A cheat! S = p [0, 1] V : q(p) > 0.

33 11 / 16 Preserving feasibility: a proof of existential LLL Recall that q(p) = S = p v indep. I ( p v ) v I { } p [0, 1] V : q(p) > 0 = Pr[set A v is monochromatic] (with cheating)

34 11 / 16 Preserving feasibility: a proof of existential LLL Recall that q(p) = S = indep. I ( p v ) v I { } p [0, 1] V : q(p) > 0 (with cheating) p v = Pr[set A v is monochromatic] Let x u = Pr[element u is red]. Then p v = x u + (1 x u ). u set A }{{ v u set A }} v {{} Pr[A v all red] Pr[A v all blue]

35 11 / 16 Preserving feasibility: a proof of existential LLL Recall that q(p) = S = indep. I ( p v ) v I { } p [0, 1] V : q(p) > 0 (with cheating) p v = Pr[set A v is monochromatic] Let x u = Pr[element u is red]. Then p v = x u + (1 x u ). u set A v u set A }{{ v } multilinear in x

36 11 / 16 Preserving feasibility: a proof of existential LLL Recall that q(p) = S = indep. I ( p v ) v I { } p [0, 1] V : q(p) > 0 (with cheating) p v = Pr[set A v is monochromatic] Let x u = Pr[element u is red]. Then p v = x u + (1 x u ). u set A v u set A }{{ v } multilinear in x Recall: the sets containing element u form a clique. = the vertices { v : p v depends on x u } form a clique.

37 11 / 16 Preserving feasibility: a proof of existential LLL Recall that q(p) = S = indep. I ( p v ) v I { } p [0, 1] V : q(p) > 0 (with cheating) p v = Pr[set A v is monochromatic] Let x u = Pr[element u is red]. Then p v = x u + (1 x u ). u set A v u set A }{{ v } multilinear in x Recall: the sets containing element u form a clique. = the vertices { v : p v depends on x u } form a clique. Easy Fact: ind set clique 1. So each monomial v I ( p v) in q is multilinear in x.

38 11 / 16 Preserving feasibility: a proof of existential LLL Recall that q(p) = S = indep. I ( p v ) v I { } p [0, 1] V : q(p) > 0 (with cheating) p v = Pr[set A v is monochromatic] Let x u = Pr[element u is red]. Then p v = x u + (1 x u ). u set A v u set A }{{ v } multilinear in x Recall: the sets containing element u form a clique. = the vertices { v : p v depends on x u } form a clique. Easy Fact: ind set clique 1. So each monomial v I ( p v) in q is multilinear in x. Rounding: Either x u 0 or x u 1 will increase q(p), and preserve feasibility.

39 11 / 16 Preserving feasibility: a proof of existential LLL Recall that q(p) = S = indep. I ( p v ) v I { } p [0, 1] V : q(p) > 0 (with cheating) p v = Pr[set A v is monochromatic] Let x u = Pr[element u is red]. Then p v = x u + (1 x u ). u set A v u set A }{{ v } multilinear in x Recall: the sets containing element u form a clique. = the vertices { v : p v depends on x u } form a clique. Easy Fact: ind set clique 1. So each monomial v I ( p v) in q is multilinear in x. Rounding: Either x u 0 or x u 1 will increase q(p), and preserve feasibility. Conclusion: Final coloring x is integral, and p S, so p = 0 (e.g., by Shearer s lemma).

40 A rounding approach to LLL 1: procedure LLLBYROUNDING 2: Initial fractional coloring: each element 1 2 red and 1 2 blue 3: for i 1,..., m do 4: Color element i red or blue, whichever preserves feasibility in S 5: return integer coloring Required to prove: For existential LLL: coloring element either red or blue preserves feasibility in S For algorithmic LLL: can efficiently decide feasibility for S 12 / 16

41 A rounding approach to LLL 1: procedure LLLBYROUNDING 2: Initial fractional coloring: each element 1 2 red and 1 2 blue 3: for i 1,..., m do 4: Color element i red or blue, whichever preserves feasibility in S 5: return integer coloring Required to prove: For existential LLL: coloring element either red or blue preserves feasibility in S For algorithmic LLL: can efficiently decide feasibility for S 12 / 16

42 Main algorithmic results 13 / 16 Let G have n = V and maximum degree d. Let α (0, 1). Theorem (The FPTAS) Suppose that (1 + α)p S. There is a deterministic algorithm ( n giving a (1 + ɛ) approximation to q(p) in time ɛα ) O(log(d)/ α). Theorem (Subexponential LLL) Suppose there are m variables and event probabilities p satisfy (1 + α)p S. There is a deterministic algorithm constructing a point in ( n nm ) O( m/α log d). i=1 E i in time α

43 14 / 16 Hard core model and counting weighted independent sets Define univariate partition function Z (p) = p I = q( p1), for p C. indep. I I [Weitz 04] [Sly 10] λ d e d FPTAS NP-hard R

44 14 / 16 Hard core model and counting weighted independent sets Define univariate partition function Z (p) = p I = q( p1), for p C. Recall: λ G = min { p : Z (p) = 0 }. indep. I λ d := inf G λ G over all graphs of maximum degree d. I [Our work] [Galanis et al. 17] [Weitz 04] [Sly 10] λ G λ λ d λ d G λ d 1 e ed d NP-hard FPTAS FPTAS NP-hard R

45 14 / 16 Hard core model and counting weighted independent sets Define univariate partition function Z (p) = p I = q( p1), for p C. Recall: λ G = min { p : Z (p) = 0 }. indep. I λ d := inf G λ G over all graphs of maximum degree d. I [Our work] [Galanis et al. 17] [Weitz 04] [Sly 10] λ G λ λ d λ d G λ d 1 e ed d NP-hard FPTAS FPTAS NP-hard R Weitz s technique: correlation decay. Our work is a multivariate, complex generalization.

46 Conclusions 15 / 16 Polynomial techniques yield a (subexponential) algorithm for the LLL. Another example where non-constructive proofs relate to stability of polynomials. Continuing phenomenon: threshold of computational hardness is boundary of zero-free region.

47 15 / 16 Conclusions Polynomial techniques yield a (subexponential) algorithm for the LLL. Another example where non-constructive proofs relate to stability of polynomials. Continuing phenomenon: threshold of computational hardness is boundary of zero-free region. Questions: FPTAS is polytime but LLL algorithm is subexponential. Improve this? Is there an FPTAS with runtime O(n/αɛ) log d? Maybe using MCMC methods?

48 Main algorithmic results 16 / 16 Let G have n = V and maximum degree d. Let α (0, 1). Theorem (Subexponential LLL) Suppose there are m variables and event probabilities p satisfy (1 + α)p S. There is a deterministic algorithm constructing a point in ( n nm ) O( m/α log d). i=1 E i in time α Previous randomized algorithms run in time poly(n/α). [Kolipaka-Szegedy 11] Deterministic algorithms with much stronger hypotheses run in time poly(n). [Chandrasekaran-Goyal-Haeupler 13]