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The concept of \emph{data depth} in non-parametric multivariate descriptivestatistics is the generalization of the univariate rank method to multivariatedata. \emph{Halfspace depth} is a measure of data depth. Given a set $S$ ofpoints and a point $p$, the halfspace depth (or rank) of $p$ is defined as theminimum number of points of $S$ contained in any closed halfspace with $p$ onits boundary. Computing halfspace depth is NP-hard, and it is equivalent to theMaximum Feasible Subsystem problem. In this paper a mixed integer program isformulated with the big-$M$ method for the halfspace depth problem. We suggesta branch and cut algorithm for these integer programs. In this algorithm,Chinneck's heuristic algorithm is used to find an upper bound and a relatedtechnique based on sensitivity analysis is used for branching. IrreducibleInfeasible Subsystem (IIS) hitting set cuts are applied. We also suggest abinary search algorithm which may be more numerically stable. The algorithmsare implemented with the BCP framework from the \textbf{COIN-OR} project.

A Branch and Cut Algorithm for the Halfspace Depth Problem