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Complexity and Algorithms for Finding a Subset of Vectors with the Longest Sum

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Computing and Combinatorics (COCOON 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10392))

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Abstract

The problem is, given a set of n vectors in a d-dimensional normed space, find a subset with the largest length of the sum vector. We prove that the problem is APX-hard for any \(\ell _p\) norm, \(p\in [1,\infty )\). For the general problem, we propose an algorithm with running time \(O(n^{d-1}(d+\log n))\), improving previously known algorithms. In particular, the two-dimensional problem can be solved in a nearly linear time. We also present an improved algorithm for the cardinality-constrained version of the problem.

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Notes

  1. 1.

    According to [12], the running time of this algorithm is as pointed above and not \(O(n^{d-1})\) as is asserted in [11].

References

  1. Baburin, A.E., Gimadi, E.K., Glebov, N.I., Pyatkin, A.V.: The problem of finding a subset of vectors with the maximum total weight. J. Appl. Industr. Math. 2(1), 32–38 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baburin, A.E., Pyatkin, A.V.: Polynomial algorithms for solving the vector sum problem. J. Appl. Industr. Math. 1(3), 268–272 (2007)

    Article  MATH  Google Scholar 

  3. Dolgushev, A.V., Kel’manov, A.V., Shenmaier, V.V.: Polynomial-time approximation scheme for a problem of partitioning a finite set into two clusters. Proc. Steklov Inst. Math. 295(Suppl 1), 47–56 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Edelsbrunner, H., O’Rourke, J., Seidel, R.: Constructing arrangements of lines and hyperplanes with applications. SIAM J. Comput. 15(2), 341–363 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gimadi, E.K., Kel’manov, A.V., Kel’manova, M.A., Khamidullin, S.A.: A posteriori detecting a quasiperiodic fragment in a numerical sequence. Pattern Recogn. Image Anal. 18(1), 30–42 (2008)

    Article  MATH  Google Scholar 

  6. Håstad, J.: Some optimal inapproximability results. J. ACM 48(4), 798–859 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Holmerin, J., Khot, S.: A new PCP outer verifier with applications to homogeneous linear equations and max-bisection. In: 36th Annual ACM Symposium on Theory of Computing, pp. 11–20. ACM, New York (2004)

    Google Scholar 

  8. Hwang, F.K., Onn, S., Rothblum, U.G.: A polynomial time algorithm for shaped partition problems. SIAM J. Optim. 10(1), 70–81 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Khot, S., Kindler, G., Mossel, E., O’Donnell, R.: Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? SIAM J. Comput. 37(1), 319–357 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Matoušek, J.: Lectures on Discrete Geometry. Springer, New York (2002)

    Book  MATH  Google Scholar 

  11. Onn, S., Schulman, L.J.: The vector partition problem for convex objective functions. Math. Oper. Res. 26(3), 583–590 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Onn, S.: Personal communication, November 2016

    Google Scholar 

  13. Pyatkin, A.V.: On the complexity of the maximum sum length vectors subset choice problem. J. Appl. Industr. Math. 4(4), 549–552 (2010)

    Article  Google Scholar 

  14. Shenmaier, V.V.: Solving some vector subset problems by Voronoi diagrams. J. Appl. Industr. Math. 10(4), 560–566 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

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

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, 4 Koptyug Avenue, 630090, Novosibirsk, Russia

    Vladimir Shenmaier

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  1. Vladimir Shenmaier
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Correspondence to Vladimir Shenmaier .

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Editors and Affiliations

  1. Department of Computing, Hong Kong Polytechnic University, Hong Kong, China

    Yixin Cao

  2. Texas A&M University, College Station, Texas, USA

    Jianer Chen

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Shenmaier, V. (2017). Complexity and Algorithms for Finding a Subset of Vectors with the Longest Sum. In: Cao, Y., Chen, J. (eds) Computing and Combinatorics. COCOON 2017. Lecture Notes in Computer Science(), vol 10392. Springer, Cham. https://doi.org/10.1007/978-3-319-62389-4_39

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  • DOI: https://doi.org/10.1007/978-3-319-62389-4_39

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-62388-7

  • Online ISBN: 978-3-319-62389-4

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