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
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Boston (1974)
Barthe, F., Guédon, O., Mendelson, S., Naor, A.: A probabilistic approach to the geometry of the \(l_p^n\)-ball. Ann. Probab. 33(2), 480–513 (2005)
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. Ind. Math. 2(1), 32–38 (2008)
Baburin, A.E., Pyatkin, A.V.: Polynomial algorithms for solving the vector sum problem. J. Appl. Ind. Math. 1(3), 268–272 (2007)
Bonnet, É., Paschos, V.T., Sikora, F.: Parameterized exact and approximation algorithms for maximum \(k\)-set cover and related satisfiability problems. RAIRO-Theor. Inform. Appl. 50(4), 227–240 (2016)
Cai, L.: Parameterized complexity of cardinality constrained optimization problems. Comput. J. 51(1), 102–121 (2008)
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)
Feige, U.: A threshold of \(\ln n\) for approximating set cover. J. ACM 45(4), 634–652 (1998)
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)
Gimadi, E., Rykov, I.: Efficient randomized algorithm for a vector subset problem. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) DOOR 2016. LNCS, vol. 9869, pp. 148–158. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-44914-2_12
Håstad, J.: Some optimal inapproximability results. J. ACM 48(4), 798–859 (2001)
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)
Hwang, F.K., Onn, S., Rothblum, U.G.: A polynomial time algorithm for shaped partition problems. SIAM J. Optim. 10(1), 70–81 (1999)
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)
Lovász, L., Simonovits, M.: Random walks in a convex body and an improved volume algorithm. Random Struct. Algorithms 4(4), 359–412 (1993)
Onn, S., Schulman, L.J.: The vector partition problem for convex objective functions. Math. Oper. Res. 26(3), 583–590 (2001)
Onn, S.: Personal communication, November 2016
Paouris, G., Valettas, P., Zinn, J.: Random version of Dvoretzky’s theorem in \(\ell _p^n\). Stochast. Process. Appl. 127(10), 3187–3227 (2017)
Pyatkin, A.V.: On the complexity of the maximum sum length vectors subset choice problem. J. Appl. Ind. Math. 4(4), 549–552 (2010)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Shenmaier, V.V.: Solving some vector subset problems by Voronoi diagrams. J. Appl. Ind. Math. 10(4), 560–566 (2016). https://doi.org/10.1134/S199047891604013X
Shenmaier, V.: Complexity and algorithms for finding a subset of vectors with the longest sum. In: Cao, Y., Chen, J. (eds.) COCOON 2017. LNCS, vol. 10392, pp. 469–480. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-62389-4_39
Shenmaier, V.V.: An exact algorithm for finding a vector subset with the longest sum. J. Appl. Ind. Math. 11(4), 584–593 (2017). https://doi.org/10.1134/S1990478917040160
Acknowledgments
This work is supported by the Russian Science Foundation under grant 16-11-10041.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Shenmaier, V. (2018). Complexity and Approximation of the Longest Vector Sum Problem. In: Solis-Oba, R., Fleischer, R. (eds) Approximation and Online Algorithms. WAOA 2017. Lecture Notes in Computer Science(), vol 10787. Springer, Cham. https://doi.org/10.1007/978-3-319-89441-6_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-89441-6_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-89440-9
Online ISBN: 978-3-319-89441-6
eBook Packages: Computer ScienceComputer Science (R0)