Norm statistics and the complexity of clustering problems

  • Authors:

  • Beth Novick

  • Affiliations:

  • Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975, United States

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2009

Quantified Score

Hi-index 0.04

Visualization

Abstract

In this paper we introduce a new class of clustering problems. These are similar to certain classical problems but involve a novel combination of @?"p-statistics and @?"q norms. We discuss a real world application in which the case p=2 and q=1 arises in a natural way. We show that, even for one dimension, such problems are NP-hard, which is surprising because the same 1-dimensional problems for the 'pure' @?"2-statistic and @?"2 norm are known to satisfy a 'string property' and can be solved in polynomial time. We generalize the string property for the case p=q. The string property need not hold when q@?p-1 and we show that instances may be constructed, for which the best solution satisfying the string property does arbitrarily poorly. We state some open problems and conjectures.