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: λ | I | . ∑ Z G ( λ ) = I ⊆ V ( G ) I independent The (random cluster formulation of the) Tutte polynomial x k ( A ) y | A | , ∑ T G ( x , y ) = A ⊆ E ( G ) 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 = ∑ 1 φ : V ( G ) → [ k ] φ is a proper coloring 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 = ∑ 1 φ : V ( G ) → [ k ] φ is a proper coloring φ : V ( G ) → [ k ] ∏ ∑ = 1 { φ ( u ) � = φ ( v ) } . uv ∈ E ( 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 = ∑ 1 φ : V ( G ) → [ k ] φ is a proper coloring φ : V ( G ) → [ k ] ∏ ∑ = 1 { φ ( u ) � = φ ( v ) } . uv ∈ E ( G ) φ : V ( G ) → [ k ] ∏ ∑ = A ( K k ) φ ( u ) , φ ( v ) . uv ∈ E ( 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 = ∑ 1 φ : V ( G ) → [ k ] φ is a proper coloring φ : V ( G ) → [ k ] ∏ ∑ = 1 { φ ( u ) � = φ ( v ) } . uv ∈ E ( G ) φ : V ( G ) → [ k ] ∏ ∑ = A ( K k ) φ ( u ) , φ ( v ) . uv ∈ E ( G ) For a symmetric k × k -matrix A (a vertex-coloring model ) define: φ : V ( G ) → [ k ] ∏ ∑ p G ( A ) = A φ ( u ) , φ ( v ) . uv ∈ E ( G ) 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 : φ : V ( G ) → [ k ] ∏ ∑ p G ( A ) = A φ ( u ) , φ ( v ) , uv ∈ E ( 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 : φ : V ( G ) → [ k ] ∏ ∑ p G ( A ) = A φ ( u ) , φ ( v ) , uv ∈ E ( G ) To go from vertex-coloring model to edge-coloring model just flip edges and vertices: φ : E ( G ) → [ k ] ∏ ∑ p G ( ? ) = ? , 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 : φ : V ( G ) → [ k ] ∏ ∑ p G ( A ) = A φ ( u ) , φ ( v ) , uv ∈ E ( G ) To go from vertex-coloring model to edge-coloring model just flip edges and vertices: φ : E ( G ) → [ k ] ∏ ∑ p G ( ? ) = ? , v ∈ V ( G ) Call a map h : N k → C an edge-coloring model and define φ : E ( G ) → [ k ] ∏ ∑ p G ( h ) = h ( φ ( δ ( v ))) . v ∈ V ( G ) 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 Z G ( λ ) for 0 ≤ λ < λ c (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 | A i , 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 Z G ( λ ) for λ ∈ C with | λ | < λ ∗ , Evaluations of the independent set polynomial Z G ( λ ) for a claw-free graph G for λ ∈ C with arg ( λ ) bounded away from − π , Evaluations of the Tutte polynomial T ( x , y 0 ) for y 0 ∈ C fixed x ∈ C with | x | ≥ C for some constant C = C ( ∆ , y 0 ) , Partition function of complex vertex-coloring models A with | A i , 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 ∗ one has Z G ( 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 ∗ one has Z G ( 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 k = 0 f ( k ) ( 0 ) z k T m ( z ) = ∑ m k ! . Then for m = O ( ln ( n / ε )) we have that | f ( z ) − T m ( 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 T m of ln ( Z G ) can efficiently be derived from the coefficients of Z G . • The k th coefficient of Z G 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 λ α ( G ) −| I | . ˆ ∑ Z G ( λ ) = I ⊂ V I independent Let ( ζ 1 , . . . , ζ α ) be the roots of ˆ Z G and define the power sums α ζ k ∑ p k = i i = 1 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 λ α ( G ) −| I | . ˆ ∑ Z G ( λ ) = I ⊂ V I independent Let ( ζ 1 , . . . , ζ α ) be the roots of ˆ Z G and define the power sums α ζ k ∑ p k = i i = 1 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 λ α ( G ) −| I | . ˆ ∑ Z G ( λ ) = I ⊂ V I independent Let ( ζ 1 , . . . , ζ α ) be the roots of ˆ Z G and define the power sums α ζ k ∑ p k = i i = 1 Surprisingly: the power sums for k = O ( log ( n / ε )) can be computed efficiently for bounded degree graphs! p 1 = | V ( G ) | ; p 2 = 2 | E ( G ) | + | V ( G ) | ; p 3 = .... 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
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