A Branch and Cut Algorithm for the Halfspace Depth Problem
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.
Cite
@article{arxiv.0705.1956,
title = {A Branch and Cut Algorithm for the Halfspace Depth Problem},
author = {Dan Chen},
journal= {arXiv preprint arXiv:0705.1956},
year = {2007}
}