Polynomial-time approximate counting and uniform sampling of contingency tables with arbitrary margins

FPRAS manuscript: one randomized algorithm counts nonnegative integer matrices with prescribed row sums, column sums and entry bounds $b_{ij}$ (zeros allowed) within relative error $\varepsilon$ with probability $1-\delta$, using unbiased bits and a polynomial bound in the input length, $\varepsilon^{-1}$ and $\log\delta^{-1}$ on every execution. Sampling manuscript: for ordinary tables, an almost-uniform sampler (TV error $2^{-k}$) in worst-case polynomial bit time and an exactly uniform sampler in expected polynomial bit time in $m,n,\log(N+1)$. Exponents are deliberately large; the results are complexity classifications, not practical algorithms, and the exact sampler is polynomial only in expectation.

FPRAS manuscript: one randomized algorithm counts nonnegative integer matrices with prescribed row sums, column sums and entry bounds $b_{ij}$ (zeros allowed) within relative error $\varepsilon$ with probability $1-\delta$, using unbiased bits and a polynomial bound in the input length, $\varepsilon^{-1}$ and $\log\delta^{-1}$ on every execution. Sampling manuscript: for ordinary tables, an almost-uniform sampler (TV error $2^{-k}$) in worst-case polynomial bit time and an exactly uniform sampler in expected polynomial bit time in $m,n,\log(N+1)$. Exponents are deliberately large; the results are complexity classifications, not practical algorithms, and the exact sampler is polynomial only in expectation.