Deterministic approximation algorithms for partition ...hladky/GraphLimits/GuusBohemia.pdf ·...

Deterministic approximation algorithms for partitionfunctions and zeros of graph polynomials

with applications to graph limits

Guus Regts

University of Amsterdam

Workshop Graph limits in Bohemian Switzerland 2018

March 25-30, 2018

Guus Regts (University of Amsterdam) Deterministic approximation algorithms for partition functions and zeros of graph polynomials1 / 37

Counting problems

number of matchings, perfect matchings, independent sets,k-colorings, the permanent, the weight enumerator of a code,partition function of Ising/Potts model, . . .

Two approaches: Monte Carlo Markov Chains (MCMC), Correlationdecay

This lecture: recent new approach due to Barvinok. Based onlocation of complex zeros.

Counting problems

number of matchings, perfect matchings, independent sets,k-colorings, the permanent, the weight enumerator of a code,partition function of Ising/Potts model, . . .Two approaches: Monte Carlo Markov Chains (MCMC), Correlationdecay

Counting problems

number of matchings, perfect matchings, independent sets,k-colorings, the permanent, the weight enumerator of a code,partition function of Ising/Potts model, . . .Two approaches: Monte Carlo Markov Chains (MCMC), Correlationdecay

Two concrete examples

The permanent of a matrix A ∈ Cn×n,

per(A) := ∑π∈Sn

∏i=1

Ai ,π(i),

and the independence polynomial of a graph G = (V ,E ),

ZG (z) := ∑I⊆VI ind.

z |I | =α(G )

∑k=0

ikzk .

Two concrete examples

The permanent of a matrix A ∈ Cn×n,

∏i=1

Ai ,π(i),

and the independence polynomial of a graph G = (V ,E ),

ZG (z) := ∑I⊆VI ind.

z |I | =α(G )

∑k=0

ikzk .

Overview

Lecture 1-2.

- approximating the permanent: what is known?- Barvinok’s approach for approximating the permanent- Taylor approximations- Zero-free regions for the permanent

Lecture 2-3

- approximating evaluations of the independence polynomial: what isknown?

- Applying Barvinok’s approach- Zeros-free regions for the independence polynomial- connections with complex dynamics

Lecture 3-4

- computing the coefficients of the independence polynomial faster- connections with sparse graph limits

Approximating the permanent: what do we want?

∏i=1

Ai ,π(i)

If per(A) > 0, A ≥ 0 and α > 1, an α-approximation to per(A) is anumber ξ > 0 such that

α−1 ≤ per(A)

ξ≤ α

We want, given α > 1, an efficient algorithm that computes anα-approximation to per(A).

If A ≥ 0, there is a randomized algorithm for computing an(1 + ε)-approximation for each ε > 0 in polynomial time by Jerrum,Sinclair and Vigoda 2004 (using MCMC).If A has values in 0, 1 and the underlying graph is an expander,there exists a (deterministic) algorithm that computes an(1 + ε)n-approximation for each ε > 0 in polynomial time, due toGamarnik and Katz, 2010 (using correlation decay approach).

∏i=1

Ai ,π(i)

α−1 ≤ per(A)

ξ≤ α

If A ≥ 0, there is a randomized algorithm for computing an(1 + ε)-approximation for each ε > 0 in polynomial time by Jerrum,Sinclair and Vigoda 2004 (using MCMC).

If A has values in 0, 1 and the underlying graph is an expander,there exists a (deterministic) algorithm that computes an(1 + ε)n-approximation for each ε > 0 in polynomial time, due toGamarnik and Katz, 2010 (using correlation decay approach).

∏i=1

Ai ,π(i)

α−1 ≤ per(A)

ξ≤ α

If A ≥ 0, there is a randomized algorithm for computing an(1 + ε)-approximation for each ε > 0 in polynomial time by Jerrum,Sinclair and Vigoda 2004 (using MCMC).If A has values in 0, 1 and the underlying graph is an expander,there exists a (deterministic) algorithm that computes an(1 + ε)n-approximation for each ε > 0 in polynomial time, due toGamarnik and Katz, 2010 (using correlation decay approach).

Barvinok’s results

Theorem (Barvinok)

Fix any positive δ such that δ < 1/2. Then there exists a constant γ > 0such that for any ε > 0 and any matrix A satisfying

|Ai ,j − 1| < δ for all i , j

one can compute in time nγ(log n−log ε) a number ξ such that

| log(per(A))− ξ| ≤ ε.

Barvinok’s results

Theorem (Barvinok)

Fix any 0 ≤ δ < 1. Then there exists a constant γ > 0 such that for anyε > 0 and any real matrix A satisfying

|1− Ai ,j | ≤ δ for all i , j

one can compute in time nγ(log n−log ε) a number ξ such that

| log(per(A))− ξ| ≤ ε.

Barvinok’s approach

Define a univariate polynomial

p(z) := per(J + z(A− J)).

Then p(0) = per(J) = n! and p(1) = per(A).

Define f (z) = log(p(z)) and approximate f (1) by writing downtruncated Taylor series of f around 0.

Need: p(z) 6= 0 for some disk centered at 0 of radius larger than 1.

Theorem (Barvinok 2016)

per(A) 6= 0 as long as each entry of A has distance at most 1/2 to 1.

Compute Taylor coefficients and analyze running time.

p(z) := per(J + z(A− J)).

Taylor coefficients I

(Taylor)

f (z) ∼= f (0) +m

∑k≥1

f (k)(0)

How to compute f (k)(0)? What should m be?

Write p(z) = ∑di=0 aiz

i = ad ∏di=1(z − ζi ). Then

f (z) = log(ad ) +d

∑i=1

log(z − ζi )

and consequently,

f ′(z) =d

∑i=1

z − ζiand thus f (k)(0) = −(k − 1)!

∑i=1

(Taylor)

f (z) ∼= f (0) +m

∑k≥1

f (k)(0)

f (z) = log(ad ) +d

∑i=1

log(z − ζi )

and consequently,

f ′(z) =d

∑i=1

z − ζiand thus f (k)(0) = −(k − 1)!

∑i=1

(Taylor)

f (z) ∼= f (0) +m

∑k≥1

f (k)(0)

f (z) = log(ad ) +d

∑i=1

log(z − ζi )

and consequently,

f ′(z) =d

∑i=1

z − ζiand thus f (k)(0) = −(k − 1)!

∑i=1

(Taylor)

f (z) ∼= f (0) +m

∑k≥1

f (k)(0)

f (z) = log(ad ) +d

∑i=1

log(z − ζi )

and consequently,

f ′(z) =d

∑i=1

z − ζi

and thus f (k)(0) = −(k − 1)!d

∑i=1

(Taylor)

f (z) ∼= f (0) +m

∑k≥1

f (k)(0)

f (z) = log(ad ) +d

∑i=1

log(z − ζi )

and consequently,

f ′(z) =d

∑i=1

z − ζiand thus f (k)(0) = −(k − 1)!

∑i=1

Taylor coefficients II

How to compute pk := ∑di=1

(Newton identities:)

k · ak = −k−1

∑i=0

ai · pk−i .

So only need to know the coefficients of p(z) = per(J + z(A− J)).Coefficient of zk :

∑π∈Sn

∏i=1

(1 + z(Ai ,π(i) − 1)) = ∑π∈Sn

∑j=0

∑U⊆[n]|U |=j

z j ∏i∈U

(Ai ,π(i) − 1).

∑j=0

z j ∑U⊆[n]|U |=j

∑π:U →[n]

(n− j)! ∏i∈U

(Ai ,π(i) − 1).

So coefficient of zk can be computed in time nO(k).

k · ak = −k−1

∑i=0

ai · pk−i .

So only need to know the coefficients of p(z) = per(J + z(A− J)).

Coefficient of zk :

∑π∈Sn

∏i=1

(1 + z(Ai ,π(i) − 1)) = ∑π∈Sn

∑j=0

∑U⊆[n]|U |=j

z j ∏i∈U

(Ai ,π(i) − 1).

∑j=0

z j ∑U⊆[n]|U |=j

∑π:U →[n]

(n− j)! ∏i∈U

(Ai ,π(i) − 1).

k · ak = −k−1

∑i=0

ai · pk−i .

∑π∈Sn

∏i=1

(1 + z(Ai ,π(i) − 1)) = ∑π∈Sn

∑j=0

∑U⊆[n]|U |=j

z j ∏i∈U

(Ai ,π(i) − 1).

∑j=0

z j ∑U⊆[n]|U |=j

∑π:U →[n]

(n− j)! ∏i∈U

(Ai ,π(i) − 1).

k · ak = −k−1

∑i=0

ai · pk−i .

∑π∈Sn

∏i=1

(1 + z(Ai ,π(i) − 1)) = ∑π∈Sn

∑j=0

∑U⊆[n]|U |=j

z j ∏i∈U

(Ai ,π(i) − 1).

∑j=0

z j ∑U⊆[n]|U |=j

∑π:U →[n]

(n− j)! ∏i∈U

(Ai ,π(i) − 1).

So coefficient of zk can be computed in time nO(k).Guus Regts (University of Amsterdam) Deterministic approximation algorithms for partition functions and zeros of graph polynomials10 / 37

Running time

Taylor series

f (z) ∼= f (0) +m

∑k≥1

f (k)(0)

k !zk = f (0)−

∑k≥1

∑i=1

zkζ−ki

If |z | < |ζi | for each i , then for m = O(log(d/ε)) we haveε-approximation.

Since 1 < |ζi | for all i and since d ≤ n we see that algorithm runs intime nO(log(n/ε)).

Running time

Taylor series

f (z) ∼= f (0) +m

∑k≥1

f (k)(0)

k !zk = f (0)−

∑k≥1

∑i=1

zkζ−ki

If |z | < |ζi | for each i , then for m = O(log(d/ε)) we haveε-approximation.

Since 1 < |ζi | for all i and since d ≤ n we see that algorithm runs intime nO(log(n/ε)).

Summary of Barvinok’s approach

Goal: approximate a polynomial p at some λ ∈ C.

Recipe

Show that p(z) is nonzero in some open (simply connected) region Ωof C that contains λ.

approximate the logarithm of p on Ω by a low order Taylorpolynomial T .

Compute the coefficients of T from the first coefficients of p.

Result: a quasi-polynomial time algorithm.

Zero-free region for the permanent

Lemma (Barvinok)

Let u1, . . . , un ∈ R2 be non-zero vectors. Suppose the angle between anypair does not exceed α for some 0 ≤ α < 2π/3.

(i) Let u = ∑ni=1 ui . Then

‖u‖ ≥ cos(α/2)n

∑i=1

‖ui‖.

(ii) Let 0 ≤ δ < cos(α/2) and let ai , bi (i ∈ [n]) be complex numberssuch that

|1− ai | < δ and |1− bi | < δ.

∑i=1

aiui and w =n

∑i=1

biui are nonzero,

and their angle does not exceed 2 arcsin(

δcos(α/2)

Zero-free region for the permanent

Lemma (Barvinok)

Let u1, . . . , un ∈ R2 be non-zero vectors. Suppose the angle between anypair does not exceed α for some 0 ≤ α < 2π/3.

(i) Let u = ∑ni=1 ui . Then

‖u‖ ≥ cos(α/2)n

∑i=1

‖ui‖.

(ii) Let 0 ≤ δ < cos(α/2) and let ai , bi (i ∈ [n]) be complex numberssuch that

|1− ai | < δ and |1− bi | < δ.

∑i=1

aiui and w =n

∑i=1

biui are nonzero,

and their angle does not exceed 2 arcsin(

δcos(α/2)

Independence polynomial

For a graph G = (V ,E ), the independence polynomial is defined as

ZG (λ) = ∑I⊆V

I independent

=α(G )

∑k=0

ikλk .

Independence polynomial

For a graph G = (V ,E ), the independence polynomial is defined as

ZG (λ) = ∑I⊆V

I independent

λ|I | = λ|I | =α(G )

∑k=0

ikλk .

ZG (1) equal the number of independent sets of G .

ZG is know as the hardcore model in statistical physics.

for p ∈ (0, 1), ZG (−p) is related to the Lovasz Local Lemma.

What is known about computing ZG

Computing ZG (1) is #P-hard when ∆(G ) ≥ 3 (Dyer and Greenhill2000).

There exist several efficient approximation algorithms based onMarkov chains for graphs of max. degree at most ∆ whenever0 ≤ λ ≤ 2/(∆− 2) (Dyer and Greenhill 2000, Vigoda 2001).

Weitz (2006) breakthrough result: a deterministic efficientapproximation algorithm for ZG (λ) whenever

0 ≤ λ < λc(∆) :=(∆− 1)∆−1

(∆− 2)∆

and ∆(G ) ≤ ∆. Correlation decay method.

Sly and Sun (2012): NP-hard to approximate ZG (λ) whenλ > λc(∆) and ∆(G ) = ∆.

0 ≤ λ < λc(∆) :=(∆− 1)∆−1

(∆− 2)∆

0 ≤ λ < λc(∆) :=(∆− 1)∆−1

(∆− 2)∆

0 ≤ λ < λc(∆) :=(∆− 1)∆−1

(∆− 2)∆

0 ≤ λ < λc(∆) :=(∆− 1)∆−1

(∆− 2)∆

Phase transition in approximate counting

λc0 ≤ λ < λc

Weitz ’06

λc < λ

Sly and Sun ’12

sub critical phase

super critical phase

Barvinok’s approach for approximating ZG

Goal: to approximate ZG at λ.

Recipe

Show that ZG (λ) is nonzero in some open (simply connected) regionΩ of C that contains λ.

approximate the logarithm of ZG on Ω by a low order Taylorpolynomial T .

Compute the coefficients of T from the first coefficients of ZG .

Theorem (Patel, R. 17)

Can compute the number of independent sets of size k in a graph G ofmax. degree ∆ in time ∆O(k)n.

Corollary

Polynomial time algorithms for bounded degree graphs.

Recipe

Corollary

Recipe

Corollary

Recipe

Corollary

Zero-free regions for ZG

For ∆ ∈N≥3 Let

λ∗(∆) :=(∆− 1)∆−1

∆∆ and λc(∆) :=(∆− 1)∆−1

(∆− 2)∆ .

Theorem (Shearer 1985)

For any graph G with max. degree at most ∆ and any λ ∈ C with|λ| ≤ λ∗(∆),

ZG (λ) 6= 0.

Theorem (Peters, R. 2017)

There exists an open region D∆ in C containing [0, λc(∆)) such that forany graph G of max. degree at most ∆ and λ ∈ D∆,

ZG (λ) 6= 0.

λ∗(∆) :=(∆− 1)∆−1

∆∆ and λc(∆) :=(∆− 1)∆−1

(∆− 2)∆ .

ZG (λ) 6= 0.

λ∗(∆) :=(∆− 1)∆−1

∆∆ and λc(∆) :=(∆− 1)∆−1

(∆− 2)∆ .

ZG (λ) 6= 0.

New algorithmic results for ZG

Theorem (Patel, R. 2017 and Harvey, Srivastava and Vondrak 2018)

Let ∆ ∈N≥3 and let λ be such that

|λ| < λ∗(∆) or 0 ≤ λ < λc(∆)

Then for each ε > 0 have a poly-time algorithm for computing ξ such that

| log(ZG (λ))− ξ| ≤ ε

for all graphs G of max. degree at most ∆.

Theorem (Bezakova, Galanis, Goldberg and Sefankovic, 2018)

#P-hard to approximate ZG (λ) when λ < −λ∗(∆) and G of max. degree∆.

New algorithmic results for ZG

Theorem (Patel, R. 2017 and Harvey, Srivastava and Vondrak 2018)

Let ∆ ∈N≥3 and let λ be such that

|λ| < λ∗(∆) or 0 ≤ λ < λc(∆)

Then for each ε > 0 have a poly-time algorithm for computing ξ such that

| log(ZG (λ))− ξ| ≤ ε

for all graphs G of max. degree at most ∆.

Theorem (Bezakova, Galanis, Goldberg and Sefankovic, 2018)

#P-hard to approximate ZG (λ) when λ < −λ∗(∆) and G of max. degree∆.

Phase transitions in approximate counting

−λ∗ λc

A useful reformulation

Fundamental recurrence for ZG : for a fixed vertex v

ZG (λ) = λZG\N [v ](λ) + ZG−v (λ).

Definition

Let us define, assuming ZG−v (λ) 6= 0,

RG ,v :=λZG\N [v ](λ)

ZG−v (λ).

A useful observation:

RG ,v 6= −1 if and only if ZG (λ) 6= 0.

A useful reformulation

Fundamental recurrence for ZG : for a fixed vertex v

ZG (λ) = λZG\N [v ](λ) + ZG−v (λ).

Definition

Let us define, assuming ZG−v (λ) 6= 0,

RG ,v :=λZG\N [v ](λ)

ZG−v (λ).

A useful observation:

RG ,v 6= −1 if and only if ZG (λ) 6= 0.

A recurrence relation

Definition

Let H be a graph with fixed vertex u0. Let u1, . . . , ud be the neighbors ofu0 in H (in any order). Set H0 = H − v0 and define for i = 1, . . . , d ,Hi := Hi−1 − ui . Then Hd = H \N [u0].

Suppose ZHi(λ) 6= 0 for all i = 0, . . . , d . Then

RH,u0 =λ

∏di=1(1 + RHi−1,ui )

Proof sketch of Shearer’s bound

Let G = (V ,E ) be of max. degree at most ∆ and let λ be such that

|λ| ≤ λ∗(∆) := (∆−1)∆−1

∆∆ . Then ZG (λ) 6= 0.

Proof.

Idea: assume G connected. Use the identity

RH,u0 =λ

∏di=1(1 + RHi−1,ui )

to prove inductively that the following holds for all U ⊆ V \ v0 (forsome fixed v0):

(i) ZG [U ](λ) 6= 0,

(ii) if u0 ∈ U has a neighbour in V \ U, then |RG [U ],u0| < 1/∆.

Proof sketch of Shearer’s bound

Let G = (V ,E ) be of max. degree at most ∆ and let λ be such that

|λ| ≤ λ∗(∆) := (∆−1)∆−1

∆∆ . Then ZG (λ) 6= 0.

Proof.

Idea: assume G connected. Use the identity

RH,u0 =λ

∏di=1(1 + RHi−1,ui )

to prove inductively that the following holds for all U ⊆ V \ v0 (forsome fixed v0):

(i) ZG [U ](λ) 6= 0,

(ii) if u0 ∈ U has a neighbour in V \ U, then |RG [U ],u0| < 1/∆.

Proof of Sokal’s conjecture: ideas

There exists an open region D∆ in C containing [0, λc(∆)) such that forany graph G of max. degree at most ∆ and λ ∈ D∆, ZG (λ) 6= 0.

F (x1, . . . , xd ) = λ/d

∏i=1

(1 + xi ).

Find a ‘trapping region’ for F . Not straightforward, univariate versionf (x) = λ/(1 + x)d is not a contraction on R≥0.

2 Find conjugation g = ϕ f ϕ−1 with |g ′| < 1 on R≥0.

3 Find open sets D∆ in the parameter space and ‘trapping region’ Usuch that for all z ∈ U and λ ∈ D∆, g(z) ∈ U .

4 Show that U also works for ϕ F ϕ−1.

F (x1, . . . , xd ) = λ/d

∏i=1

(1 + xi ).

Find a ‘trapping region’ for F .

Not straightforward, univariate versionf (x) = λ/(1 + x)d is not a contraction on R≥0.

F (x1, . . . , xd ) = λ/d

∏i=1

(1 + xi ).

F (x1, . . . , xd ) = λ/d

∏i=1

(1 + xi ).

F (x1, . . . , xd ) = λ/d

∏i=1

(1 + xi ).

F (x1, . . . , xd ) = λ/d

∏i=1

(1 + xi ).

Regular trees and complex dynamics

Fix ∆ ≥ 3, Let Tk be the tree with k levels, in which each non-leaf vertexhas ∆− 1 descendants. Then

RTk ,v =λ

(1 + RTk−1,v )∆−1= f k(λ)

λ(α) :=

−α(∆− 1)∆−1

(∆− 1 + α)∆

∣∣∣|α| < 1

λ(1) = −λ∗ and λ(−1) = λc .

(i) Let λ ∈ U. Then ZTk(λ) 6= 0 for all k .

(ii) For any λ ∈ ∂U, there exists λ′ arbitrarily close to λ and k ∈N suchthat ZTk

(λ′) = 0.

RTk ,v =λ

(1 + RTk−1,v )∆−1= f k(λ)

λ(α) :=

−α(∆− 1)∆−1

(∆− 1 + α)∆

∣∣∣|α| < 1

λ(1) = −λ∗ and λ(−1) = λc .

(λ′) = 0.

RTk ,v =λ

(1 + RTk−1,v )∆−1= f k(λ)

λ(α) :=

−α(∆− 1)∆−1

(∆− 1 + α)∆

∣∣∣|α| < 1

λ(1) = −λ∗ and λ(−1) = λc .

(λ′) = 0.

RTk ,v =λ

(1 + RTk−1,v )∆−1= f k(λ)

λ(α) :=

−α(∆− 1)∆−1

(∆− 1 + α)∆

∣∣∣|α| < 1

λ(1) = −λ∗ and λ(−1) = λc .

(λ′) = 0.

RTk ,v =λ

(1 + RTk−1,v )∆−1= f k(λ)

λ(α) :=

−α(∆− 1)∆−1

(∆− 1 + α)∆

∣∣∣|α| < 1

λ(1) = −λ∗ and λ(−1) = λc .

(λ′) = 0.

Regular trees and complex dynamics II

λ(α) :=

−α(∆− 1)∆−1

(∆− 1 + α)∆

∣∣∣|α| < 1

Theorem (Bezakova, Galanis, Goldberg, Stefankovic 2018)

Let λ /∈ U and suppose λ is not positive. Then it is #P-hard toapproximate ZG (λ) on graphs G of max. degree ∆.

Question

Is it true that for any λ ∈ U and any graph G of max. degree at most ∆,ZG (λ) 6= 0?

Regular trees and complex dynamics II

λ(α) :=

−α(∆− 1)∆−1

(∆− 1 + α)∆

∣∣∣|α| < 1

Theorem (Bezakova, Galanis, Goldberg, Stefankovic 2018)

Let λ /∈ U and suppose λ is not positive. Then it is #P-hard toapproximate ZG (λ) on graphs G of max. degree ∆.

Question

Is it true that for any λ ∈ U and any graph G of max. degree at most ∆,ZG (λ) 6= 0?

Recipe

1 Show that p(z) is nonzero in some open (simply connected) region Ωof C that contains λ.

2 approximate the logarithm of p on Ω by a low order Taylorpolynomial T .

3 Compute the coefficients of T from the first coefficients of p.

In many cases, item 3 can be done in polynomial time!

Recipe

1 Show that p(z) is nonzero in some open (simply connected) region Ωof C that contains λ.

2 approximate the logarithm of p on Ω by a low order Taylorpolynomial T .

3 Compute the coefficients of T from the first coefficients of p.

In many cases, item 3 can be done in polynomial time!

Polynomial time algorithms from Barvinok’s approach

The matching polynomial (Patel, R. 2017; zero-free region byHeilman and Lieb 1972)

Partition functions of edge-coloring models/Holants (Patel, R. 2017;zero-free region by R. 2016+)

Tutte polynomial (Patel, R. 2017; zero-free region by Jackson,Procacci and Sokal 2013)

Partition functions of spin models (Patel, R. 2017; zero-free region byBarvinok and Soberon 2017)

Hypergraph Ising model (Liu, Sinclair, Srivastava 2017)

Weight enumerator for linear codes (Barvinok and R. 2017+)

The chromatic polynomial

Let χG denote the chromatic polynomial of a graph G .

Theorem (Jackson, Procacci and Sokal 2013)

For any graph G of max. degree at most ∆ and any z such that|z | ≥ 6.91∆,

χG (z) 6= 0.

Question

Is it true that for any ∆ > 2 there exists ε > 0 such that for any graph Gof max. degree at most ∆,

χG (z) 6= 0

whenever <(z) > ∆ and |=(z)|a ≤ ε?

Counting induced subgraphs

Definition

Let H and G be graphs. Then ind(H,G ) denotes the number of subsetsS ⊆ V (G ) such that H = G [S ].

Example

Let H be the graph consisting of k isolated vertices. Then ind(H,G ) is #independent sets of size k .

Theorem (Patel, R. 2017+)

Let ∆ ∈N. Then there is a ∆O(k)n-time algorithm that computesind(H,G ) for graphs H and G of order k and n respectively and maxdegree at most ∆.

Definition

Example

Definition

Example

Application to the independence polynomial and questions

Corollary

Let ∆ ∈N. We can compute the number of independent sets of size k ina graph G of maximum degree at most ∆ and order n in ∆O(k)n time.

Theorem (Nederlof, Patel, R. 2018+)

There exists a constant c > such that we can compute the number ofindependent sets of size k in a planar graph G of order n in cO(k)n time

Questions

Does there exist c > 0 and an algorithm, which given a graph H oforder k and a planar graph G of order n that computes ind(H,G ) intime cO(k)poly(n)?

Recall p(z) = per(J + z(A− J)). Does there exist c > 0 and analgorithm that computes the coefficient of zk of p in timecO(k)poly(n) for matrices A of order n?

Application to the independence polynomial and questions

Corollary

Let ∆ ∈N. We can compute the number of independent sets of size k ina graph G of maximum degree at most ∆ and order n in ∆O(k)n time.

Theorem (Nederlof, Patel, R. 2018+)

There exists a constant c > such that we can compute the number ofindependent sets of size k in a planar graph G of order n in cO(k)n time

Questions

Does there exist c > 0 and an algorithm, which given a graph H oforder k and a planar graph G of order n that computes ind(H,G ) intime cO(k)poly(n)?

Recall p(z) = per(J + z(A− J)). Does there exist c > 0 and analgorithm that computes the coefficient of zk of p in timecO(k)poly(n) for matrices A of order n?

Computing # ind. sets of size k : Tools

If H connected, then ind(H,G ) can be computed in time ∆(G )kn.

Definition

Call a graph parameter f additive if f (G1 ∪ G2) = f (G1) + f (G2).

Lemma (Csikvari and Frenkel 2016)

Let f = ∑ni=1 ai ind(Hi , ·) be a graph parameter. Then f is additive if and

only if ai = 0 whenever Hi is not connected.

Definition

Computing # ind. sets of size k : Tools II

Let ζ1, . . . ζα be the roots of ZG (λ) = ∑αk=0 ikλk and let pk = ∑α

i=1 ζ−ki .

The pk are additive graph parameters.

(Newton identities)

k · ik = −k−1

∑m=0

im · pk−m ⇐⇒ pk = −k−1

∑m=1

im · pk−m − k · ik

Corollary

We can write

pk(·) =m

∑i=1

ak,i · ind(Hi , ·)

for certain coefficients ai and connected graphs Hi of order at most k .

Use this to iteratively compute the coefficients ai for each pk .

i=1 ζ−ki .

(Newton identities)

k · ik = −k−1

∑m=0

im · pk−m ⇐⇒ pk = −k−1

∑m=1

Corollary

We can write

pk(·) =m

∑i=1

Use this to iteratively compute the coefficients ai for each pk .

i=1 ζ−ki .

(Newton identities)

k · ik = −k−1

∑m=0

im · pk−m ⇐⇒ pk = −k−1

∑m=1

Corollary

We can write

pk(·) =m

∑i=1

Use this to iteratively compute the coefficients ai for each pk .Guus Regts (University of Amsterdam) Deterministic approximation algorithms for partition functions and zeros of graph polynomials34 / 37

Connections with sparse graph limits

Definition

A sequence of bounded degree graphs (Gn) is called Benjamini-Schrammconvergent if |V (Gn)| → ∞ and for each connected graph H the sequence

ind(H,Gn)

|V (Gn)|

is convergent.

(Observation)

If (Gn) is convergent, |∆(Gn)| ≤ ∆ and |λ| < (∆−1)∆−1

∆∆ , then the sequence

log(ZGn(λ))

|V (Gn)|

is convergent.

Announcement: Workshop Algorithmic and Combinatorial Aspects ofPartition Functions, in Amsterdam, August 23-24.

https://sites.google.com/site/guusregts/home/workshop

Thank you for your attention!

Announcement: Workshop Algorithmic and Combinatorial Aspects ofPartition Functions, in Amsterdam, August 23-24.

https://sites.google.com/site/guusregts/home/workshop

Thank you for your attention!

Some references

Barvinok, A. (2016). Combinatorics and complexity of partition functions (Vol. 274). Springer.

Barvinok, A. & Regts, G. (2017). Weighted counting of non-negative integer points in a subspace. arXiv preprintarXiv:1706.05423.

Bezakova, I., Galanis, A., Goldberg, L. A., & Stefankovic, D. (2017). Inapproximability of the independent setpolynomial in the complex plane. STOC 2018, arXiv preprint arXiv:1711.00282.

Liu, J., Sinclair, A., & Srivastava, P. (2017). The Ising partition function: Zeros and deterministic approximation.FOCS 2017, arXiv preprint arXiv:1704.06493.

Csikvari, P., & Frenkel, P. E. (2016). Benjamini-Schramm continuity of root moments of graph polynomials. EuropeanJournal of Combinatorics, 52, 302-320.

Patel, V., & Regts, G. (2017). Deterministic polynomial-time approximation algorithms for partition functions andgraph polynomials. SIAM Journal on Computing, 46(6), 1893-1919.

Patel, V., & Regts, G. (2017). Computing the number of induced copies of a fixed graph in a bounded degree graph.arXiv preprint arXiv:1707.05186.

Peters, H., & Regts, G. (2017). On a conjecture of Sokal concerning roots of the independence polynomial. arXivpreprint arXiv:1701.08049.

Regts, G. (2017). Zero-free regions of partition functions with applications to algorithms and graph limits.Combinatorica, 1-29.

Sly, A., & Sun, N. (2012). The computational hardness of counting in two-spin models on d-regular graphs. FOCS 2012(pp. 361–369).

Weitz, D. (2006). Counting independent sets up to the tree threshold. STOC 2016 (pp. 140–149).

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