An approximation algorithm for multidimensional assignment problems minimizing the sum of squared errors |
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Authors: | Yusuke Kuroki Tomomi Matsui |
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Institution: | a Graduate School of Information Science and Technology, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan b Department of Information and System Engineering, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan |
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Abstract: | Given a complete k-partite graph G=(V1,V2,…,Vk;E) satisfying |V1|=|V2|=?=|Vk|=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 Qd 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. |
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Keywords: | Multidimensional assignment problem Approximation algorithm Second-order cone programming Data-association problem Data fusion |
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