Distances between optimal solutions of mixed-integer programs

A classic result of Cook et al. (Math. Program. 34:251–264, 1986) bounds the distances between optimal solutions of mixed-integer linear programs and optimal solutions of the corresponding linear relaxations. Their bound is given in terms of the number of variables and a parameter Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \varDelta $$\end{document}, which quantifies sub-determinants of the underlying linear inequalities. We show that this distance can be bounded in terms of Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \varDelta $$\end{document} and the number of integer variables rather than the total number of variables. To this end, we make use of a result by Olson (J. Number Theory 1:8–10, 1969) in additive combinatorics and demonstrate how it implies feasibility of certain mixed-integer linear programs. We conjecture that our bound can be improved to a function that only depends on Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \varDelta $$\end{document}, in general.