Advertisement

Hardness, approximability, and fixed-parameter tractability of the clustered shortest-path tree problem

  • Mattia D’EmidioEmail author
  • Luca Forlizzi
  • Daniele Frigioni
  • Stefano Leucci
  • Guido Proietti
Article
  • 26 Downloads

Abstract

Given an n-vertex non-negatively real-weighted graph G, whose vertices are partitioned into a set of k clusters, a clustered network design problem on G consists of solving a given network design optimization problem on G, subject to some additional constraints on its clusters. In particular, we focus on the classic problem of designing a single-source shortest-path tree, and we analyse its computational hardness when in a feasible solution each cluster is required to form a subtree. We first study the unweighted case, and prove that the problem is \({\textsf {NP}}\)-hard. However, on the positive side, we show the existence of an approximation algorithm whose quality essentially depends on few parameters, but which remarkably is an O(1)-approximation when the largest out of all the diameters of the clusters is either O(1) or \(\varTheta (n)\). Furthermore, we also show that the problem is fixed-parameter tractable with respect to k or to the number of vertices that belong to clusters of size at least 2. Then, we focus on the weighted case, and show that the problem can be approximated within a tight factor of O(n), and that it is fixed-parameter tractable as well. Finally, we analyse the unweighted single-pair shortest path problem, and we show it is hard to approximate within a (tight) factor of \(n^{1-\epsilon }\), for any \(\epsilon >0\).

Keywords

Clustered shortest-path tree problem Hardness Approximation algorithms Fixed-parameter tractability Network design 

Notes

References

  1. Bao X, Liu Z (2012) An improved approximation algorithm for the clustered traveling salesman problem. Inf Process Lett 112(23):908–910MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bilò D, Grandoni F, Gualà L, Leucci S, Proietti G (2015) Improved purely additive fault-tolerant spanners. In: Proceedings 23rd European symposium on algorithms (ESA), volume 9294 of Lecture notes in computer science. Springer, pp 167–178Google Scholar
  3. Björklund A, Husfeldt T, Kaski P, Koivisto M (2007) Fourier meets möbius: fast subset convolution. In: Proceedings 39th ACM symposium on theory of computing (STOC). ACM, pp 67–74Google Scholar
  4. Byrka J, Grandoni F, Rothvoß T, Sanità L (2013) Steiner tree approximation via iterative randomized rounding. J ACM 60(1):6MathSciNetCrossRefzbMATHGoogle Scholar
  5. Dasgupta S, Papadimitriou CH, Vazirani U (2008) Algorithms, 1st edn. McGraw-Hill Inc, New YorkGoogle Scholar
  6. D’Emidio M, Forlizzi L, Frigioni D, Leucci S, Proietti G (2016) On the clustered shortest-path tree problem. In: Proceedings 17th Italian conference on theoretical computer science (ICTCS), volume 1720 of CEUR workshop proceedings, pp 263–268Google Scholar
  7. Fareed MS, Javaid N, Akbar M, Rehman S, Qasim U, Khan ZA (2012) Optimal number of cluster head selection for efficient distribution of sources in WSNs. In: Proceedings seventh international conference on broadband, wireless computing, communication and applications. IEEE, pp 626–631Google Scholar
  8. Feremans C, Labbé M, Laporte G (2003) Generalized network design problems. Eur J Oper Res 148(1):1–13MathSciNetCrossRefzbMATHGoogle Scholar
  9. Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. W. H. Freeman & Co, New YorkzbMATHGoogle Scholar
  10. Garg N, Konjevod G, Ravi R (2000) A polylogarithmic approximation algorithm for the group Steiner tree problem. J Algorithms 37(1):66–84MathSciNetCrossRefzbMATHGoogle Scholar
  11. Guttmann-Beck N, Hassin R, Khuller S, Raghavachari B (2000) Approximation algorithms with bounded performance guarantees for the clustered traveling salesman problem. Algorithmica 28(4):422–437MathSciNetCrossRefzbMATHGoogle Scholar
  12. Halperin E, Krauthgamer R (2003) Polylogarithmic inapproximability. In: Proceedings 35th ACM symposium on theory of computing (STOC), pp 585–594Google Scholar
  13. Lin C, Wu BY (2016) On the minimum routing cost clustered tree problem. J Comb Optim 31(1):1–16MathSciNetCrossRefGoogle Scholar
  14. Sevgi C, Kocyigit A (2008) On determining cluster size of randomly deployed heterogeneous WSNs. IEEE Commun Lett 12(4):232–234CrossRefGoogle Scholar
  15. Wu BY, Lancia G, Bafna V, Chao K-M, Ravi R, Tang CY (1998) A polynomial time approximation scheme for minimum routing cost spanning trees. In: Proceedings 9th ACM-SIAM symposium on discrete algorithms (SODA), pp 21–32Google Scholar
  16. Wu BY, Lin C (2015) On the clustered Steiner tree problem. J Comb Optim 30(2):370–386MathSciNetCrossRefzbMATHGoogle Scholar
  17. Zou P, Li H, Wang W, Xin C, Zhu B (2018) Finding disjoint dense clubs in a social network. Theor Comput Sci 734:15–23MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Information Engineering, Computer Science and MathematicsUniversity of L’AquilaL’AquilaItaly
  2. 2.Department of Computer ScienceETH ZürichZürichSwitzerland
  3. 3.Istituto di Analisi dei Sistemi e Informatica “Antonio Ruberti” Consiglio Nazionale delle RicercheRomeItaly

Personalised recommendations