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Complexity and Approximation of the Longest Vector Sum Problem

  • Vladimir ShenmaierEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10787)

Abstract

Given a set of n vectors in a d-dimensional normed space, consider the problem of finding a subset with the largest length of the sum vector. We prove that, for any \(\ell _p\) norm, \(p\in [1,\infty )\), the problem is hard to approximate within a factor better than \(\min \{\alpha ^{1/p},\sqrt{\alpha }\}\), where \(\alpha =16{\text{/ }}17\). In the general case, we show that the cardinality-constrained version of the problem is hard for approximation factors better than \(1-1/e\) and is W[2]-hard with respect to the cardinality of the solution. For both original and cardinality-constrained problems, we propose a randomized \((1-\varepsilon )\)-approximation algorithm that runs in polynomial time when the dimension of space is \(O(\log n)\). The algorithm has a linear time complexity for any fixed d and \(\varepsilon \in (0,1)\).

Keywords

Computational geometry Vector sum Normed space Inapproximability bound W[.]-hardness Approximation algorithm 

Notes

Acknowledgments

This work is supported by the Russian Science Foundation under grant 16-11-10041.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia

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