Smoothed counting of 0–1 points in polyhedra

Abstract

Given a system of linear equations ℓifalse(xfalse)=βi$$ {\ell}_i(x)={\beta}_i $$ in an n$$ n $$‐vector x$$ x $$ of 0–1 variables, we compute the expectation of exp{}prefix−∑iγi()ℓifalse(xfalse)prefix−βi2$$ \exp \left\{-{\sum}_i{\gamma}_i{\left({\ell}_i(x)-{\beta}_i\right)}^2\right\} $$, where x$$ x $$ is a vector of independent Bernoulli random variables and γi>0$$ {\gamma}_i>0 $$ are constants. The algorithm runs in quasi‐polynomial nOfalse(lnnfalse)$$ {n}^{O\left(\ln n\right)} $$ time under some sparseness condition on the matrix of the system. The result is based on the absence of the zeros of the analytic continuation of the expectation for complex probabilities, which can also be interpreted as the absence of a phase transition in the Ising model with a sufficiently strong external field. We discuss applications to perfect matchings in hypergraphs and randomized rounding in discrete optimization.

More on zeros and approximation of the Ising partition function

“…If H is not regular, one can choose instead the maximum entropy distribution, which also ensures that the expected number of selected edges containing any given vertex is exactly 1, while the weights of all perfect matchings remain equal. The maximum entropy distribution exists if an only if there exists a positive fractional perfect matching, that is, an assignment of positive real weights to the edges of H such that for every vertex of H the sum of weights of the edges containing it is exactly 1, see [Ba23] for details.…”

“…Proof. The inequality of Part (1) is proved, for example, in [Ba23], see Lemma 6.1 there. To prove the inequality of Part (2), let f (α) = sin(τ α) and g(α) = τ sin α.…”