A Branch and Cut Algorithm for the Halfspace Depth Problem

TL;DR

This paper introduces a branch and cut algorithm for efficiently computing the halfspace depth of points in multivariate data, addressing the NP-hardness of the problem with innovative optimization techniques.

Contribution

It formulates a mixed integer program for the halfspace depth problem and develops a novel branch and cut algorithm using heuristics and IIS cuts.

Findings

Algorithm effectively computes halfspace depth for complex datasets.

Uses Chinneck's heuristic for upper bounds and IIS cuts for optimization.

Implemented within the COIN-OR BCP framework.

Abstract

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