On the Integrality Gap of Binary Integer Programs with Gaussian Data

For a binary integer program (IP) \(\max c^\mathsf {T}x, Ax \le b, x \in \{0,1\}^n\), where \(A \mathbb {R}^{m \times n}\) and \(c {R}^n\) have independent Gaussian entries the right-hand side \(b {R}^m\) satisfies that its negative coordinates \(\ell _2\) norm at most n/10, we prove gap between value of linear programming relaxation IP is upper bounded by \(\text {poly}(m)(\log n)^2 / n\) with probability least \(1-1/n^7-2^{-\text {poly}(m)}\). Our results give analogue classical integrality result Dyer Frieze (Math. O.R., 1989) in case random packing IPs. In constrast to case, our depends only polynomially on m instead exponentially. By recent breakthrough work Dey, Dubey Molinaro (SODA, 2021), bound immediately implies branch requires \(n^{\text {poly}(m)}\) time IPs good probability, which polynomial when number constraints fixed.