Counting perfect matchings via random matrices Mark Rudelson based on joint works with Alex Samorodnitky and Ofer Zeitouni Department of Mathematics University of Michigan Mark Rudelson (Michigan) Counting perfect matchings via random matrices 1 / 1 A perfect matching is a permutation π 2 Πn such that (j; π(j) 2 E for all j 2 [n]. #(perfect matchings) = per(A); where A is the adjacency matrix of the graph. Perfect matchings Let Γ = (L; R; V) be an n × n bipartite graph. Mark Rudelson (Michigan) Counting perfect matchings via random matrices 2 / 1 #(perfect matchings) = per(A); where A is the adjacency matrix of the graph. π = (1; 3; 2; 4; 6; 5) Perfect matchings 1 1 2 2 Let Γ = (L; R; V) be an n × n bipartite graph. A perfect matching is a permutation 3 3 π 2 Πn such that (j; π(j) 2 E for all j 2 [n]. 4 4 5 5 6 6 π = (2; 1; 6; 4; 5; 3) Mark Rudelson (Michigan) Counting perfect matchings via random matrices 2 / 1 #(perfect matchings) = per(A); where A is the adjacency matrix of the graph. π = (2; 1; 6; 4; 5; 3) Perfect matchings 1 1 2 2 Let Γ = (L; R; V) be an n × n bipartite graph. A perfect matching is a permutation 3 3 π 2 Πn such that (j; π(j) 2 E for all j 2 [n]. 4 4 5 5 6 6 π = (1; 3; 2; 4; 6; 5) Mark Rudelson (Michigan) Counting perfect matchings via random matrices 2 / 1 π = (2; 1; 6; 4; 5; 3) π = (1; 3; 2; 4; 6; 5) Perfect matchings 1 1 2 2 Let Γ = (L; R; V) be an n × n bipartite graph. A perfect matching is a permutation 3 3 π 2 Πn such that (j; π(j) 2 E for all j 2 [n]. 4 4 #(perfect matchings) = per(A); 5 5 where A is the adjacency matrix of the graph. 6 6 Mark Rudelson (Michigan) Counting perfect matchings via random matrices 2 / 1 Evaluation of determinants is fast: use e.g., triangularization by Gaussian elimination. Coppersmith-Winograd algorithm: Running time: O(n2:376). Determinant of A: n X Y det(A) = sign(π) aj,π(j): π2Πn j=1 Evaluation of permanents is #P-complete (Valiant 1979) Permanent of a matrix Let A be an n × n matrix with ai;j ≥ 0. Permanent of A: n X Y per(A) = aj,π(j): π2Πn j=1 Mark Rudelson (Michigan) Counting perfect matchings via random matrices 3 / 1 Evaluation of determinants is fast: use e.g., triangularization by Gaussian elimination. Evaluation of permanents is Coppersmith-Winograd algorithm: #P-complete (Valiant 1979) Running time: O(n2:376). Permanent of a matrix Let A be an n × n matrix with ai;j ≥ 0. Permanent of A: Determinant of A: n n X Y X Y per(A) = aj,π(j): det(A) = sign(π) aj,π(j): π2Πn j=1 π2Πn j=1 Mark Rudelson (Michigan) Counting perfect matchings via random matrices 3 / 1 Evaluation of permanents is #P-complete (Valiant 1979) Permanent of a matrix Let A be an n × n matrix with ai;j ≥ 0. Permanent of A: Determinant of A: n n X Y X Y per(A) = aj,π(j): det(A) = sign(π) aj,π(j): π2Πn j=1 π2Πn j=1 Evaluation of determinants is fast: use e.g., triangularization by Gaussian elimination. Coppersmith-Winograd algorithm: Running time: O(n2:376). Mark Rudelson (Michigan) Counting perfect matchings via random matrices 3 / 1 Permanent of a matrix Let A be an n × n matrix with ai;j ≥ 0. Permanent of A: Determinant of A: n n X Y X Y per(A) = aj,π(j): det(A) = sign(π) aj,π(j): π2Πn j=1 π2Πn j=1 Evaluation of determinants is fast: use e.g., triangularization by Gaussian elimination. Evaluation of permanents is Coppersmith-Winograd algorithm: #P-complete (Valiant 1979) Running time: O(n2:376). Mark Rudelson (Michigan) Counting perfect matchings via random matrices 3 / 1 Qn Qn 0 0 per(A) = i=1 di · j=1 dj · per(A ) Deterministic bounds Linial–Samorodnitsky–Wigderson algoritm: if per(A) > 0, then one can find in polynomial time diagonal matrices D; D0 such that the renormalized matrix A0 = D0AD is almost doubly stochastic: n X 0 1 − " < ai;j < 1 + "; for all j = 1;:::; n i=1 n X 0 1 − " < ai;j < 1 + "; for all i = 1;:::; n j=1 Mark Rudelson (Michigan) Counting perfect matchings via random matrices 4 / 1 Deterministic bounds Linial–Samorodnitsky–Wigderson algoritm: if per(A) > 0, then one can find in polynomial time diagonal matrices D; D0 such that the renormalized matrix A0 = D0AD is almost doubly stochastic: n X 0 1 − " < ai;j < 1 + "; for all j = 1;:::; n i=1 n X 0 1 − " < ai;j < 1 + "; for all i = 1;:::; n j=1 Qn Qn 0 0 per(A) = i=1 di · j=1 dj · per(A ) Mark Rudelson (Michigan) Counting perfect matchings via random matrices 4 / 1 Gurvits-Samorodnitsky estimator (2014) + Linial–Samorodnitsky–Wigderson algorithm yields the multiplicative error 2n. Deterministic bounds Linial–Samorodnitsky–Wigderson algoritm: reduces permanent estimates to almost doubly stochastic matrices Van der Waerden conjecture, proved by Falikman and Egorychev: if A is doubly stochastic, then n! 1 ≥ per(A) ≥ ≈ e−n nn Linial–Samorodnitsky–Wigderson algorithm estimates the permanent with the multiplicative error at most en Mark Rudelson (Michigan) Counting perfect matchings via random matrices 5 / 1 Deterministic bounds Linial–Samorodnitsky–Wigderson algoritm: reduces permanent estimates to almost doubly stochastic matrices Van der Waerden conjecture, proved by Falikman and Egorychev: if A is doubly stochastic, then n! 1 ≥ per(A) ≥ ≈ e−n nn Linial–Samorodnitsky–Wigderson algorithm estimates the permanent with the multiplicative error at most en Gurvits-Samorodnitsky estimator (2014) + Linial–Samorodnitsky–Wigderson algorithm yields the multiplicative error 2n. Mark Rudelson (Michigan) Counting perfect matchings via random matrices 5 / 1 Deficiency: running time is O(n10) (improved later to O(n7 log4 n)). Barvinok’s estimator (improves Godsil–Gutman estimator). Let A be the adjacency matrix. Let Γ be an n × n random matrix having independent N(0; aij) entries. Then 2 per(A) = E det (Γ): Estimator: per(A) ≈ det2(Γ). Advantage: Barvinok’s estimator is faster than any other algorithm. How accurate is it? Probabilistic estimates Jerrum–Sinclair–Vigoda algorithm estimates the permanent of any matrix with constant multiplicative error with high probability. Mark Rudelson (Michigan) Counting perfect matchings via random matrices 6 / 1 Barvinok’s estimator (improves Godsil–Gutman estimator). Let A be the adjacency matrix. Let Γ be an n × n random matrix having independent N(0; aij) entries. Then 2 per(A) = E det (Γ): Estimator: per(A) ≈ det2(Γ). Advantage: Barvinok’s estimator is faster than any other algorithm. How accurate is it? Probabilistic estimates Jerrum–Sinclair–Vigoda algorithm estimates the permanent of any matrix with constant multiplicative error with high probability. Deficiency: running time is O(n10) (improved later to O(n7 log4 n)). Mark Rudelson (Michigan) Counting perfect matchings via random matrices 6 / 1 Advantage: Barvinok’s estimator is faster than any other algorithm. How accurate is it? Probabilistic estimates Jerrum–Sinclair–Vigoda algorithm estimates the permanent of any matrix with constant multiplicative error with high probability. Deficiency: running time is O(n10) (improved later to O(n7 log4 n)). Barvinok’s estimator (improves Godsil–Gutman estimator). Let A be the adjacency matrix. Let Γ be an n × n random matrix having independent N(0; aij) entries. Then 2 per(A) = E det (Γ): Estimator: per(A) ≈ det2(Γ). Mark Rudelson (Michigan) Counting perfect matchings via random matrices 6 / 1 How accurate is it? Probabilistic estimates Jerrum–Sinclair–Vigoda algorithm estimates the permanent of any matrix with constant multiplicative error with high probability. Deficiency: running time is O(n10) (improved later to O(n7 log4 n)). Barvinok’s estimator (improves Godsil–Gutman estimator). Let A be the adjacency matrix. Let Γ be an n × n random matrix having independent N(0; aij) entries. Then 2 per(A) = E det (Γ): Estimator: per(A) ≈ det2(Γ). Advantage: Barvinok’s estimator is faster than any other algorithm. Mark Rudelson (Michigan) Counting perfect matchings via random matrices 6 / 1 Probabilistic estimates Jerrum–Sinclair–Vigoda algorithm estimates the permanent of any matrix with constant multiplicative error with high probability. Deficiency: running time is O(n10) (improved later to O(n7 log4 n)). Barvinok’s estimator (improves Godsil–Gutman estimator). Let A be the adjacency matrix. Let Γ be an n × n random matrix having independent N(0; aij) entries. Then 2 per(A) = E det (Γ): Estimator: per(A) ≈ det2(Γ). Advantage: Barvinok’s estimator is faster than any other algorithm. How accurate is it? Mark Rudelson (Michigan) Counting perfect matchings via random matrices 6 / 1 Identity matrix: multiplicative error at least exp(cn) w.h.p. p Matrix of all ones: multiplicative error at most exp(C log n) (Goodman, 1963). Can it be improved? Precision bounds for Barvinok’s estimator Theorem (Barvinok) Let A be anyn × n matrix with non-negative entries. Then, with probability 1 − δ, ((1 − ") · θ)n per(A) ≤ det2(Γ) ≤ C per(A); where C is an absolute constant and θ = 0:28 for real Gaussian matrices; θ = 0:56 for complex Gaussian matrices; Mark Rudelson (Michigan) Counting perfect matchings via random matrices 7 / 1 Identity matrix: multiplicative error at least exp(cn) w.h.p. p Matrix of all ones: multiplicative error at most exp(C log n) (Goodman, 1963).