Approximation algorithms for graph polynomials and partition - - PowerPoint PPT Presentation

Approximation algorithms for graph polynomials and partition functions.

Guus Regts

University of Amsterdam

14 June 2016, Dagstuhl Seminar Graph Polynomials: Towards a Comparative Theory

Based on joint work with Viresh Patel (University of Amsterdam) Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 1 / 15

1 Graph polynomials and partition functions

The independent set polynomial: ZG(λ) =

∑

I⊆V (G) I independent

λ|I|. The (random cluster formulation of the) Tutte polynomial TG(x, y) =

∑

A⊆E(G)

xk(A)y |A|, here k(A) denotes the number of components of the graph (V , A).

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 2 / 15

1 Graph polynomials and partition functions

The number of proper k-colorings of a graph G

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 3 / 15

1 Graph polynomials and partition functions

The number of proper k-colorings of a graph G = ∑

φ:V (G)→[k] φ is a proper coloring

1

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 3 / 15

1 Graph polynomials and partition functions

The number of proper k-colorings of a graph G = ∑

φ:V (G)→[k] φ is a proper coloring

1 =

∑

φ:V (G)→[k] ∏ uv∈E(G)

1{φ(u)=φ(v)}.

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 3 / 15

1 Graph polynomials and partition functions

The number of proper k-colorings of a graph G = ∑

φ:V (G)→[k] φ is a proper coloring

1 =

∑

φ:V (G)→[k] ∏ uv∈E(G)

1{φ(u)=φ(v)}. =

∑

φ:V (G)→[k] ∏ uv∈E(G)

A(Kk)φ(u),φ(v).

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 3 / 15

1 Graph polynomials and partition functions

The number of proper k-colorings of a graph G = ∑

φ:V (G)→[k] φ is a proper coloring

1 =

∑

φ:V (G)→[k] ∏ uv∈E(G)

1{φ(u)=φ(v)}. =

∑

φ:V (G)→[k] ∏ uv∈E(G)

A(Kk)φ(u),φ(v). For a symmetric k × k-matrix A (a vertex-coloring model) define: pG(A) =

∑

φ:V (G)→[k] ∏ uv∈E(G)

Aφ(u),φ(v).

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 3 / 15

1 Graph polynomials and partition functions

The partition function of the vertex-coloring model A: pG(A) =

∑

φ:V (G)→[k] ∏ uv∈E(G)

Aφ(u),φ(v),

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 4 / 15

1 Graph polynomials and partition functions

The partition function of the vertex-coloring model A: pG(A) =

∑

φ:V (G)→[k] ∏ uv∈E(G)

Aφ(u),φ(v), To go from vertex-coloring model to edge-coloring model just flip edges and vertices: pG(?) =

∑

φ:E(G)→[k] ∏ v∈V (G)

?,

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 4 / 15

1 Graph polynomials and partition functions

The partition function of the vertex-coloring model A: pG(A) =

∑

φ:V (G)→[k] ∏ uv∈E(G)

Aφ(u),φ(v), To go from vertex-coloring model to edge-coloring model just flip edges and vertices: pG(?) =

∑

φ:E(G)→[k] ∏ v∈V (G)

?, Call a map h : Nk → C an edge-coloring model and define pG(h) =

∑

φ:E(G)→[k] ∏ v∈V (G)

h(φ(δ(v))).

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 4 / 15

2 Previous work: correlation decay

Correlation decay method (assuming ∆ = ∆(G) is constant) yields an efficient deterministic approximation algorithm (FPTAS) for: Evaluations of the independent set polynomial ZG(λ) for 0 ≤ λ < λc (Weitz, 2006) The number of matchings in a graph (Bayati, Gamarnik, Katz, Nair and Tetali, 2007) The number of k-colorings of a graph for k > α∆ + 1 for α large enough (Lu and Yin, 2013) Partition function of real vertex-coloring models A with |Ai,j − 1| ≤ c/∆ (for some constant c) (Lu and Yin, 2013)

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 5 / 15

3 New approach: Taylor approximations

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 6 / 15

3 New approach: Taylor approximations

High level idea: try to approximate the logarithm of the polynomial/partition function with a low order Taylor polynomial.

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 6 / 15

3 New approach: Taylor approximations

New approach (assuming ∆ = ∆(G) is constant) yields an efficient deterministic approximation algorithm (FPTAS) for: Evaluations of the independent set polynomial ZG(λ) for λ ∈ C with |λ| < λ∗, Evaluations of the independent set polynomial ZG(λ) for a claw-free graph G for λ ∈ C with arg(λ) bounded away from −π, Evaluations of the Tutte polynomial T(x, y0) for y0 ∈ C fixed x ∈ C with |x| ≥ C for some constant C = C(∆, y0), Partition function of complex vertex-coloring models A with |Ai,j − 1| ≤ c/∆ (for some constant c), Partition function of complex edge-coloring models h with |h(α) − 1| ≤ c′/∆ (for some constant c′).

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 7 / 15

4 How does it work: independence polynomial

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 8 / 15

4 How does it work: independence polynomial

Theorem (Lov´ asz ’75, Dobrushin ’96, Shearer ’99, Scott and Sokal ’05)

For each ∆ ∈ N there exists a constant λ∗ = λ∗(∆) > 0 such that for each graph G of maximum degree at most ∆ and λ such that |λ| ≤ λ∗

• ne has

ZG(λ) = 0.

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 8 / 15

4 How does it work: independence polynomial

Theorem (Lov´ asz ’75, Dobrushin ’96, Shearer ’99, Scott and Sokal ’05)

For each ∆ ∈ N there exists a constant λ∗ = λ∗(∆) > 0 such that for each graph G of maximum degree at most ∆ and λ such that |λ| ≤ λ∗

• ne has

ZG(λ) = 0.

Lemma (Barvinok 2015)

Let p be a polynomial of degree n such that p(z) = 0 for all |z| ≤ C for some C > 0. Let f (z) = ln p(z) for |z| < C and let Tm(z) = ∑m

k=0 f (k)(0) zk k! . Then for m = O(ln(n/ε)) we have that

|f (z) − Tm(z)| ≤ ε.

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 8 / 15

4 How does it work: independence polynomial

• The coefficients of the m-th order Taylor polynomial Tm of ln(ZG) can

efficiently be derived from the coefficients of ZG.

• The kth coefficient of ZG is equal to the number of independent sets of

size k of G. How do we compute that for k = O(ln n/ε)?

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 9 / 15

4 How does it work: independence polynomial

For a graph G = (V , E) define ˆ ZG(λ) =

∑

I⊂V Iindependent

λα(G)−|I|. Let (ζ1, . . . , ζα) be the roots of ˆ ZG and define the power sums pk =

α

∑

i=1

ζk

i

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 10 / 15

4 How does it work: independence polynomial

For a graph G = (V , E) define ˆ ZG(λ) =

∑

I⊂V Iindependent

λα(G)−|I|. Let (ζ1, . . . , ζα) be the roots of ˆ ZG and define the power sums pk =

α

∑

i=1

ζk

i

Surprisingly: the power sums for k = O(log(n/ε)) can be computed efficiently for bounded degree graphs!

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 10 / 15

4 How does it work: independence polynomial

For a graph G = (V , E) define ˆ ZG(λ) =

∑

I⊂V Iindependent

λα(G)−|I|. Let (ζ1, . . . , ζα) be the roots of ˆ ZG and define the power sums pk =

α

∑

i=1

ζk

i

Surprisingly: the power sums for k = O(log(n/ε)) can be computed efficiently for bounded degree graphs! p1 = |V (G)|; p2 = 2|E(G)| + |V (G)|; p3 = ....

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 10 / 15

4 How does it work: partition functions

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 11 / 15

4 How does it work: partition functions

Theorem (Barvinok and Sober´

• n 2016)

Let G be a graph with n vertices and let A be an vertex-coloring model. If |Ai,j − 1| ≤ 0.35

∆(G) for each i, j = 1, . . . , k, then

pG(A) = 0.

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 11 / 15

4 How does it work: partition functions

Theorem (Barvinok and Sober´

• n 2016)

Let G be a graph with n vertices and let A be an vertex-coloring model. If |Ai,j − 1| ≤ 0.35

∆(G) for each i, j = 1, . . . , k, then

pG(A) = 0. Let J be the all-ones matrix and define for A such that |Ai,j − 1| < 0.34/∆(G) p(z) = pG(J + z(A − J)). Then p(0) = k|V (G)|, p(1) = pG(A), and p(z) = 0 for all z such that |z| ≤ 0.35

0.34.

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 11 / 15

5 Discussion and concluding remarks

Correlation decay method, for counting independent sets, matchings, graph coloring etc., is related to absence of phase transition, i.e., uniqueness of Gibss measure.

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 12 / 15

5 Discussion and concluding remarks

Correlation decay method, for counting independent sets, matchings, graph coloring etc., is related to absence of phase transition, i.e., uniqueness of Gibss measure. The method presented here is also related to absence of phase transition, i.e., via the Lee-Yang theorem no complex zeros ⇒ no phase transition.

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 12 / 15

5 Discussion and concluding remarks

Complexity of the independent set polynomial at positive λ: λc λ < λc Weitz ’06 λc < λ Sly and Sun ’12 sub critical phase super critical phase

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 13 / 15

5 Discussion and concluding remarks

Correlation decay method only seems to work for positive real numbers. The method presented here also works for complex numbers. New method does not work in all situations where correlation decay methods works (so far...).

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 14 / 15

Thank you for your attention!

Guus Regts (University of Amsterdam) Approximation algorithms for graph polynomials and partition functions. 15 / 15