Smoothed counting of 0-1 points in polyhedra

TL;DR

This paper introduces a quasi-polynomial time algorithm for computing smoothed counts of 0-1 solutions in polyhedra, leveraging complex analysis and phase transition absence, with applications in hypergraph matchings and optimization.

Contribution

It presents a novel quasi-polynomial algorithm for expectation computation in 0-1 polyhedra based on complex zeros analysis, extending previous methods.

Findings

Algorithm runs in $n^{O(\, ext{ln}\, n)}$ time under sparseness conditions.

Absence of zeros in the analytic continuation indicates no phase transition in related Ising models.

Applications include hypergraph matchings and randomized discrete optimization.

Abstract

Given a system of linear equations $\ell_i(x)=\beta_i$ in an $n$-vector $x$ of 0-1 variables, we compute the expectation of $\exp\left\{- \sum_i \gamma_i \left(\ell_i(x) - \beta_i\right)^2\right\}$, where $x$ is a vector of independent Bernoulli random variables and $\gamma_i >0$ are constants. The algorithm runs in quasi-polynomial $n^{O(\ln n)}$ 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.