An FPRAS for counting common bases of two matroids
Theorem 1.1: one randomized oracle algorithm, given independence oracles for rank-r matroids on [n] and rational ε,δ∈(0,1), outputs Z within relative error ε of the number of common bases with probability at least 1−δ, outputs zero when there are none, and uses polynomially many oracle calls and bit operations on every execution. Corollaries give FPRASes and almost-uniform samplers for common independent sets of prescribed, unrestricted or maximum size, even for different ranks. The method is a Jerrum-Sinclair-Vigoda style annealed chain with defect states, using the homogeneous Tutte theorem of Branden-Huh (Lorentzian polynomials). It gives no exact counting and no deterministic algorithm, and does not treat three or more matroids. The October companion gives an FPRAS for common integer bases of two polymatroids with binary capacities.
Theorem 1.1: one randomized oracle algorithm, given independence oracles for rank-r matroids on [n] and rational ε,δ∈(0,1), outputs Z within relative error ε of the number of common bases with probability at least 1−δ, outputs zero when there are none, and uses polynomially many oracle calls and bit operations on every execution. Corollaries give FPRASes and almos…
Theorem 1.1: one randomized oracle algorithm, given independence oracles for rank-r matroids on [n] and rational ε,δ∈(0,1), outputs Z within relative error ε of the number of common bases with probability at least 1−δ, outputs zero when there are none, and uses polynomially many oracle calls and bit operations on every execution. Corollaries give FPRASes and almost-uniform samplers for common independent sets of prescribed, unrestricted or maximum size, even for different ranks. The method is a Jerrum-Sinclair-Vigoda style annealed chain with defect states, using the homogeneous Tutte theorem of Branden-Huh (Lorentzian polynomials). It gives no exact counting and no deterministic algorithm, and does not treat three or more matroids. The October companion gives an FPRAS for common integer bases of two polymatroids with binary capacities.
Theorem 1.1: one randomized oracle algorithm, given independence oracles for rank-r matroids on [n] and rational ε,δ∈(0,1), outputs Z within relative error ε of the number of common bases with probability at least 1−δ, outputs zero when there are none, and uses polynomially many oracle calls and bit operations on every execution. Corollaries give FPRASes and almost-uniform samplers for common independent sets of prescribed, unrestricted or maximum size, even for different ranks. The method is a Jerrum-Sinclair-Vigoda style annealed chain with defect states, using the homogeneous Tutte theorem of Branden-Huh (Lorentzian polynomials). It gives no exact counting and no deterministic algorithm, and does not treat three or more matroids. The October companion gives an FPRAS for common integer bases of two polymatroids with binary capacities.