The Maximum Box Problem and its Application to Data Analysis

Authors:
Jonathan Eckstein;Peter L. Hammer;Ying Liu;Mikhail Nediak;Bruno Simeone
Affiliations:
Rutgers Business School and RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway, NJ 08854, USA. jeckstei@rutcor.rutgers.edu;RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway, NJ 08854, USA. hammer@rutcor.rutgers.edu;RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway, NJ 08854, USA. yingliu@rutcor.rutgers.edu;RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway, NJ 08854, USA. msnediak@rutcor.rutgers.edu;Department of Statistics, “La Sapienza” University, Piazzale Aldo Moro 5, 00185 Rome, Italy. bruno.simeone@uniroma1.it
Venue:
Computational Optimization and Applications
Year:
2002

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Abstract

Given two finite sets of points X+ and X− in \Bbb Rn, the maximum box problem consists of finding an interval (“box”) B = {x : l ≤ x ≤ u} such that B ∩ X− = ∅, and the cardinality of B ∩ X+ is maximized. A simple generalization can be obtained by instead maximizing a weighted sum of the elements of B ∩ X+. While polynomial for any fixed n, the maximum box problem is \cal{NP}-hard in general. We construct an efficient branch-and-bound algorithm for this problem and apply it to a standard problem in data analysis. We test this method on nine data sets, seven of which are drawn from the UCI standard machine learning repository.