Locations of Zeros and Approximate Counting Chihao Zhang Shanghai Jiao Tong University Apr. 11 2019
Generating Function ∞ a k x k = 1 + 4 x + 2 x 2 ∑ F IS ( x ) = k =0
̂ n ∑ F IS (1) = a k = # of independent sets k =1 # P - hard to evaluate (not easier than NP ) Approximation F ∀ ε > 0, 1 − ε < F IS (1) < 1 + ε
Technique for Approximate Counting “Sample from Gibbs Measure” Markov Chain Monte Carlo Simulate a Markov chain on all independent sets Correlation Decay Approximately compute the marginals of Gibbs measure using graph recursions
Barvinok’s Method Alexander Barvinok Brook Taylor
A bit of math… G = log F IS T m G ( z ) = First m terms of the Maclaurin series of G ) ( = G (0) + G ′ � (0) z + G (2) (0) z 2 + G (3) (0) z 3 + … + G ( m − 1) (0) ( m − 1)! z m − 1 2 3! Is T m G (1) a good approximation of G ? (for su ffi ciently small m )
Assume F ( x ) has roots α 1 , …, α n ∈ ℂ A simple calculation yields | T m G ( t ) − G ( t ) | ≤ poly( | G | ) ⋅ (1 − γ ) m | α i | if t < min i t 1 − γ = min i | α i |
Question Remains… up to m = O (log | G | ) How to evaluate T m G ( t )? For IS, trivially in | G | O ( m ) Mathematician: I don’t care ( n O (log n ) is already a good notion for e ffi cient computation) A. Barvinok Computer Scientists: We do care about poly-time Patel & Regts (2017) : Can be done in | G | ⋅ 2 O ( m ) time Viresh Patel Guus Regts
The Message (Many) generating functions are approximable in zero-free regions It is challenging to identify the location of zeros For partition functions of physical model, lots of work has been done…
Zero-free Region Lee-Yang Theorem (1952) : The partition function of the Ising model has all of its complex roots on the 1-circle T.D. Lee C.N. Yang Partition Function (Physics) A.Sokal… Generating Function (Math) D.Wagner, A.Barvinok… Graph Polynomial (CS)
Zero-free Region of IS “cardiac zero-free region” on trees [Bezakova, Galanis, Goldberg, Š tefankovi č 2018] It is NP -hard to approximate IS polynomial out of the zero-free region Cropped from “Note on the zero-free region of the hard-core model” by F . Bencs and P . Csikvári. (http://arxiv.org/abs/1807.08963)
Matchings m ∑ m k x k Matching Polynomial : F M ( x ) = k =0
Holant Problem f 1 f 2 1 2 σ ∈ {0,1} E ∏ ∑ F M (1) = # ℳ = f v ( σ ), v ∈ V 4 3 f 4 f 3 f v ( σ ) is the indicator function of the event σ chooses at most one of v ′ � s incident edges Holant : f v can be arbitrary {0,1} E ( v ) → ℂ Holant( ℱ ) : f v ∈ ℱ This talk : f v is symmetric
Dichotomy Theorems [Cai, Guo, Williams 2013] Holant( ℱ ) is either in P or # P -complete # P-complete Most problems are hard, unless it enjoys PTIME some good algebraic property
Approximate Counting Approximation: FPTAS/FPRAS for the partition function • We know a few tractable islands • We know a few intractable islands # ind. sets # matchings A dichotomy for approximating Holant( ℱ )? # edge covers
Barvinok’s Method Theorem (Barvinok 16; Patel & Regts 18 ) There exists an FPTAS to approximate F H (1) if the polynomial F H ( x ) is zero - free in a δ - strip containing [0,1] for some δ > 0 ℂ True for Matching (Heilmann-Lieb) δ General Holant? 0 1
Asano Contraction In a proof of the Lee-Yang theorem…
Holant Polynomial | E | ∑ b i ⋅ x i Recall that F H ( x ) = i =0 We obtain the polynomial by gluing small polynomials x (2) x (1) 1 3 x 1 x 3 x (1) x (2) 1 3 x 2 x (2) x (1) 2 2 F ( x 1 , x 2 , x 3 ) F ( x (1) 1 , x (2) 1 , x (1) 2 , x (2) 2 , x (1) 3 , x (2) 3 ) Asano’s contraction
Ruelle’s Bound x 1 x x 2 F ′ � ( x ) F ( x 1 , x 2 ) Ruelle’s Theorem (Ruelle 71) F ′ � ( x ) is zero - free in − ¯ A ⋅ ¯ F ( x 1 , x 2 ) is zero - free in A A
Ruelle’s Theorem F ′ � ( x ) is zero - free in − ¯ A ⋅ ¯ F ( x 1 , x 2 ) is zero - free in A A f x (2) x (1) 1 3 x 1 x 3 x (1) x (2) 1 3 f f x 2 x (2) x (1) 2 2 F ( x 1 , x 2 , x 3 ) Corollary F ( x 1 , x 2 , x 3 ) is zero - free in δ -strip of [0,1] 2 ) is zero - free in H ε p f ( x (1) 1 , x (2) H ε := { z ∈ ℂ : ℜ z > − ε }
Local Polynomial Constraint Function f = [ f 0 , f 1 , …, f d ] d k =0 ( k ) ⋅ f k ⋅ z k d ∑ Local Polynomial p f ( z ) = Identify the family of f such that p f is H ε zero - free
Second-Order Recurrence for some ( a , b , c ) ≠ (0,0,0) f = [ f 0 , f 1 , …, f d ] af k + bf k +1 + cf k +2 = 0 ∀ 0 ≤ k ≤ d − 2 f k ≥ 0 & • matchings and perfect matchings • parity • edge covers • Fibonacci gates • all ternary symmetric functions
Local Polynomial of Second-Order Recurrence The local polynomial is of the form p f ( z ) = ( p + qz ) d + ( r + sz ) d Good if it is H ε zero-free Not much information when it is not
Holographic Reduction A tweak of the local polynomial without changing the global value p ′ � p f f F H F ′ � H F H (1) = F ′ � H (1) Leslie Valiant
Holographic Reduction p f ( z ) = ( p + qz ) d + ( r + sz ) d a free w d + ( ( rw + s ) + ( r − ws ) z ) d ( ( pw + q ) + ( p − wq ) z ) • A many-to-many reduction between counting problems • Central tool in establishing dichotomy results for exact counting
Results for SO Holant In a joint work with C. Liao, H.Guo and P . Lu, we obtain approximation algorithm for SO Holant except d , λ 2 sin 2 π d , …, λ i sin i π [0, λ sin π d , …,0] d − 2 2 ,0] if d even [0,1,0, λ ,0,…,0, λ d − 1 2 ] if d odd [0,1,0, λ ,0,…,0, λ The later case is equivalent to approximating #PM #PM: central open problem in approximate counting
Ternary functions satisfy SO recurrence Approximating Holant( f ) on cubic graph is either in poly - time or equivalent to aprroximating #PM A “Dichotomy” [0,1,0, λ ] or [ λ ,0,1,0] #PM FPTAS/FPRAS otherwise
Conclusion Barvinok’s method is powerful and seems to be universal Conjecture Uniqueness of Gibbs measure (Spatial mixing) implies absence of complex zero We are more familiar with Gibbs measure… No idea how to prove absence of zero in general
Conclusion q -coloring independent sets more Holant Recent progress • IS: tree is not the worst case (arxiv 1903.05462) • Potts (q-coloring): matching the best uniqueness bound (Last week) • some progress on asymmetric Holant
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