An abstract model for branching and its application to mixed integer programming

The selection of branching variables is a key component of branch-and-bound algorithms for solving mixed-integer programming (MIP) problems since the quality of the selection procedure is likely to have a significant effect on the size of the enumeration tree. State-of-the-art procedures base the selection of variables on their “LP gains”, which is the dual bound improvement obtained after branching on a variable. There are various ways of selecting variables depending on their LP gains. However, all methods are evaluated empirically. In this paper we present a theoretical model for the selection of branching variables. It is based upon an abstraction of MIPs to a simpler setting in which it is possible to analytically evaluate the dual bound improvement of choosing a given variable. We then discuss how the analytical results can be used to choose branching variables for MIPs, and we give experimental results that demonstrate the effectiveness of the method on MIPLIB 2010 “tree” instances where we achieve a 5%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5\%$$\end{document} geometric average time and node improvement over the default rule of SCIP, a state-of-the-art MIP solver.