Stability of Networks in Stretchable Graphs

  • Davide Bilò
  • Michael Gatto
  • Luciano Gualà
  • Guido Proietti
  • Peter Widmayer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5869)

Abstract

In classic optimization theory, the concept of stability refers to the study of how much and in which way the optimal solutions of a given minimization problem Π can vary as a function of small perturbations of the input data. Motivated by congestion problems arising in shortest-path based communication networks, in this paper we restrict ourselves to the case in which Π is actually a network design problem on a given graph G = (V,E,w) of |V| = n nodes, |E| = m edges, and with a positive real weight w(e) on each edge e ∈ E. We focus on a subclass of perturbations, that we call stretching perturbations, in which the weights of the edges of G can be increased by at most a fixed multiplicative real factor λ ≥ 1.

For this class of perturbations, we address the problem of computing the stability number of any given subgraph H of G containing at least an optimal solution of Π, namely the maximum stretching factor for which H keeps on maintaining an optimal solution. Furthermore, given a stretching factor λ, we study the problem of constructing a minimal subgraph of G with stability number greater or equal to λ.

We develop a general technique to solve both problems. By applying this technique to the minimum spanning tree and the single-source shortest paths tree (SPT) problems, we obtain \({\cal O}(m\alpha(m,n))\) and \({\cal O}(mn(m+n \log n))\) time algorithms, respectively, where α(·,·) is the functional inverse of Ackermann’s function. Furthermore, for the SPT problem, we show that if H coincides with the set of all optimal solutions, then the time complexity can be reduced to \({\cal O}(mn)\). Finally, for the single-source single-destination shortest path problem, if the optimal solutions of the input instance happen to form a set of vertex-disjoint paths, and H coincides with this set, then we show that we can compute the stability number in \({\cal O}(mn + n^2 \log n)\) time.

Keywords

Communication Networks Shortest Paths Edge Perturbation Stability Theory 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beyer, H.-G., Sendhoff, B.: Robust optimization - a comprehensive survey. Computer Methods in Applied Mechanics and Engineering 196(33-34), 3190–3218 (2007)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Dixon, B., Rauch, M., Tarjan, R.E.: Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM J. Comput. 21(6), 1184–1192 (1992)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Gal, T., Greenberg, H.J. (eds.): Advances in sensitivity analysis and parametric programming. Int. Series in Operations Research and Management Science, vol. 6. Kluwer Academic Publishers, Boston (1997)MATHGoogle Scholar
  4. 4.
    Kouvelis, P., Yu, G.: Robust discrete optimization and its applications. Kluwer Academic Publishers, Dordrecht (1997)CrossRefMATHGoogle Scholar
  5. 5.
    Pettie, S., Ramachandran, V.: An optimal minimum spanning tree algorithm. J. ACM 49(1), 16–34 (2002)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Pettie, S.: Sensitivity analysis of minimum spanning trees in sub-inverse-Ackermann time. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 964–973. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Shier, D.R., Witzgall, C.: Edge tolerances in shortest path and network flow problems. Networks 10(4), 277–291 (1980)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Tarjan, R.E.: Sensitivity analysis of minimum spanning trees and shortest path trees. Inf. Proc. Letters 14(1), 30–33 (1982)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Davide Bilò
    • 1
  • Michael Gatto
    • 1
  • Luciano Gualà
    • 2
  • Guido Proietti
    • 3
    • 4
  • Peter Widmayer
    • 1
  1. 1.Institut für Theoretische InformatikETH ZurichZürichSwitzerland
  2. 2.Dipartimento di MatematicaUniversità di Tor VergataRomaItaly
  3. 3.Dipartimento di InformaticaUniversità di L’AquilaL’AquilaItaly
  4. 4.Istituto di Analisi dei Sistemi ed InformaticaCNRRomaItaly

Personalised recommendations