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Optimal Partitioning Which Maximizes the Weighted Sum of Products

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Frontiers in Algorithmics (FAW 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10336))

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Abstract

We consider the problem of partitioning n real numbers to K nonempty groups, so that the weighted sum of products over all groups is maximized. Formally, given \(S=\{r_1,\ldots ,r_n\}\) and \(W=(w_1,\ldots ,w_K)\) where \(w_i\ge 0\), we look for a partition of S into K nonempty groups \(S_1,\ldots ,S_K\), so that \(\sum _{g=1}^{K} (w_g\cdot \prod _{r_j\in S_g}r_j)\) is maximized. Our main result is an \(O(n^2)\) time algorithm for finding an optimal partition.

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Notes

  1. 1.

    According to one of the reviewers, this can be proved from “exact cover by 3-sets” problem, similar to Yao’s NP-hardness proof for the “subset product” problem.

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Authors and Affiliations

  1. University of Hong Kong, Pokfulam, Hong Kong SAR, China

    Kai Jin

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  1. Kai Jin
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Correspondence to Kai Jin .

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Editors and Affiliations

  1. University of Electronic Science and Technology of China, Chengdu, China

    Mingyu Xiao

  2. University of Bergen, Bergen, Norway

    Frances Rosamond

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Jin, K. (2017). Optimal Partitioning Which Maximizes the Weighted Sum of Products. In: Xiao, M., Rosamond, F. (eds) Frontiers in Algorithmics. FAW 2017. Lecture Notes in Computer Science(), vol 10336. Springer, Cham. https://doi.org/10.1007/978-3-319-59605-1_12

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  • DOI: https://doi.org/10.1007/978-3-319-59605-1_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-59604-4

  • Online ISBN: 978-3-319-59605-1

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