An approximation algorithm for multidimensional assignment problems minimizing the sum of squared errors

  • Authors:

  • Yusuke Kuroki;Tomomi Matsui

  • Affiliations:

  • Graduate School of Information Science and Technology, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan;Department of Information and System Engineering, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan

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

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Abstract

Given a complete k-partite graph G=(V"1,V"2,...,V"k;E) satisfying |V"1|=|V"2|=...=|V"k|=n and weights of all k-cliques of G, the k-dimensional assignment problem finds a partition of vertices of G into a set of (pairwise disjoint) n k-cliques that minimizes the sum total of weights of the chosen cliques. In this paper, we consider a case in which the weight of a clique is defined by the sum of given weights of edges induced by the clique. Additionally, we assume that vertices of G are embedded in the d-dimensional space Q^d and a weight of an edge is defined by the square of the Euclidean distance between its two endpoints. We describe that these problem instances arise from a multidimensional Gaussian model of a data-association problem. We propose a second-order cone programming relaxation of the problem and a polynomial time randomized rounding procedure. We show that the expected objective value obtained by our algorithm is bounded by (5/2-3/k) times the optimal value. Our result improves the previously known bound (4-6/k) of the approximation ratio.