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 novel optimization techniques.

Contribution

It formulates a mixed integer program and develops a branch and cut algorithm using heuristics and IIS cuts to improve computation of halfspace depth.

Findings

The proposed algorithm effectively computes halfspace depth for complex datasets.

The binary search approach offers improved numerical stability.

Implementation within the BCP framework demonstrates practical applicability.

Abstract

The concept of data depth in non-parametric multivariate descriptive statistics is the generalization of the univariate rank method to multivariate data. Halfspace depth is a measure of data depth. Given a set S of points and a point p, the halfspace depth (or rank) k 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 thesis a mixed integer program is formulated with the big-M method for the halfspace depth problem. We suggest a branch and cut algorithm. 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 stable numerically. The algorithms are implemented with the BCP framework from the COIN-OR project.