Smoothed counting of 0–1 points in polyhedra

Given a system of linear equations ℓ i ( x ) = β $$ {\ell}_i(x)={\beta}_i in an n -vector 0–1 variables, we compute the expectation exp − ∑ γ 2 \exp \left\{-{\sum}_i{\gamma}_i{\left({\ell}_i(x)-{\beta}_i\right)}^2\right\} , where is vector independent Bernoulli random variables and > 0 {\gamma}_i>0 are constants. The algorithm runs quasi-polynomial O ln {n}^{O\left(\ln n\right)} time under some sparseness condition on matrix system. result based absence zeros analytic continuation for complex probabilities, which can also be interpreted as phase transition Ising model with sufficiently strong external field. We discuss applications to perfect matchings hypergraphs randomized rounding discrete optimization.