On the Integrality Gap of Binary Integer Programs with Gaussian Data

TL;DR

This paper analyzes the integrality gap of binary integer programs with Gaussian data, showing it is small with high probability, and connects this to the efficiency of branch-and-bound algorithms.

Contribution

It provides a Gaussian analogue of classical integrality gap results and introduces a meta-theorem linking integrality gaps to branch-and-bound complexity for random logconcave IPs.

Findings

The integrality gap is bounded by poly(log n)/n with high probability.

Branch-and-bound requires polynomial time for fixed number of constraints.

The results extend classical packing IP bounds to Gaussian data setting.

Abstract

For a binary integer program (IP) ${\rm max} ~ c^\mathsf{T} x, Ax \leq b, x \in \{0,1\}^n$, where $A \in \mathbb{R}^{m \times n}$ and $c \in \mathbb{R}^n$ have independent Gaussian entries and the right-hand side $b \in \mathbb{R}^m$ satisfies that its negative coordinates have $\ell_2$ norm at most $n/10$, we prove that the gap between the value of the linear programming relaxation and the IP is upper bounded by $\operatorname{poly}(m)(\log n)^2 / n$ with probability at least $1-2/n^7-2^{-\operatorname{poly}(m)}$. Our results give a Gaussian analogue of the classical integrality gap result of Dyer and Frieze (Math. of O.R., 1989) in the case of random packing IPs. In constrast to the packing case, our integrality gap depends only polynomially on $m$ instead of exponentially. Building upon recent breakthrough work of Dey, Dubey and Molinaro (SODA, 2021), we show that the integrality gap implies that branch-and-bound requires $n^{\operatorname{poly}(m)}$ time on random Gaussian IPs with good probability, which is polynomial when the number of constraints $m$ is fixed. We derive this result via a novel meta-theorem, which relates the size of branch-and-bound trees and the integrality gap for random logconcave IPs.