On a Minimum Linear Classification Problem

Authors:
Bing Lu;Hongwei Du;Xiaohua Jia;Yinfeng Xu;Binhai Zhu
Affiliations:
Department of Computer Science and Enginnering, University of Minnesota, Minneapolis, USA 55455;Department of Computer Science, City University of Hong Kong, Hong Kong;Department of Computer Science, City University of Hong Kong, Hong Kong;School of Management, Xi'an Jiaotong University, China;Department of Computer Science, Montana State University, Bozeman, USA 59717
Venue:
Journal of Global Optimization
Year:
2006

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Abstract

We study the following linear classification problem in signal processing: Given a set Bof n black point and a set W of m white points in the plane (m = O(n)), compute a minimum number of lines L such that in the arrangement of L each face contain points with the same color (i.e., either all black points or all white points). We call this the Minimum Linear Classification (MLC) problem. We prove that MLC is NP-complete by a reduction from the Minimum Line Fitting (MLF) problem; moreover, a C-approximation to MLC implies a C-approximation to the MLF problem. Nevertheless, we obtain an O(log n)-factor algorithm for MLC and we also obtain an O(log Z)-factor algorithm for MLC where Z is the minimum number of disjoint axis-parallel black/white rectangles covering B and W.