On the Integrality Gap of Binary Integer Programs with Gaussian Data

Abstract

For a binary integer program (IP) $\max c^\mathsf {T}x, Ax \le b, x \in \{0,1\}^n$, 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 $\text {poly}(m)(\log n)^2 / n$ with probability at least $1-1/n^7-2^{-\text {poly}(m)}$. 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. By recent breakthrough work of Dey, Dubey and Molinaro (SODA, 2021), the bound on the integrality gap immediately implies that branch and bound requires $n^{\text {poly}(m)}$ time on random Gaussian IPs with good probability, which is polynomial when the number of constraints m is fixed.

S. Borst, D. Dadush and S. Tiwari—This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement QIP–805241).

D. Dadush and S. Huiberts—This work was done while the author was participating in a program at the Simons Institute for the Theory of Computing.