On the Integrality Gap of Binary Integer Programs with Gaussian Data
Abstract
For a binary integer program (IP) $\max c^{\mathsf T} x, Ax \leq b, x \in \{0,1\}^n$, where $A \in \mathbb{R}^{m \times n}$ and $c \in \mathbb{R}^n$ have independent Gaussian entries and the righthand 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 $11/n^71/2^{\Omega(m)}$. 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 $n^{\operatorname{poly}(m)}$ time on random Gaussian IPs with good probability, which is polynomial when the number of constraints $m$ is fixed.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 arXiv:
 arXiv:2012.08346
 Bibcode:
 2020arXiv201208346B
 Keywords:

 Mathematics  Optimization and Control;
 Computer Science  Data Structures and Algorithms