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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 \(1-2/n^7-2^{-{\text {poly}}(m)}\). Our results give a Gaussian analogue of the classical integrality gap result of Dyer and Frieze (Math OR, 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 \(n^{{\text {poly}}(m)}\) 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.

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Correspondence to Daniel Dadush.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A preliminary version of this work was published in the proceedings of the 22nd conference on Integer Programming and Combinatorial Optimization (IPCO 2021) [3].

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).

This work was done while the second and third named authors were participating in the Fall 2020 program “Probability, Geometry, and Computation in High Dimensions” at the Simons Institute for the Theory of Computing.

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Borst, S., Dadush, D., Huiberts, S. et al. On the integrality gap of binary integer programs with Gaussian data. Math. Program. (2022). https://doi.org/10.1007/s10107-022-01828-1

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  • DOI: https://doi.org/10.1007/s10107-022-01828-1

Keywords

  • Integer programming
  • Integrality gap
  • Branch-and-bound

Mathematics Subject Classification

  • 90C10