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 O(mα(m,n)) and O(mn(m+nlogn)) 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 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 O(mn+n2logn) time. Structural Information and Communication Complexity Structural Information and Communication Complexity Look Inside Other actions Reprints and Permissions Export citation About this Book Add to Papers Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn

Stability of Networks in Stretchable Graphs / Bilò, Davide; Gatto, Michael; Gualà, Luciano; Proietti, Guido; Widmayer, Peter. - 5869:(2009), pp. 100-112. (Intervento presentato al convegno 16th International Colloquium on Structural Information and Communication Complexity tenutosi a Piran, Slovenia nel May 25-27, 2009) [10.1007/978-3-642-11476-2_9].

Stability of Networks in Stretchable Graphs

BILÒ, Davide;
2009-01-01

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 O(mα(m,n)) and O(mn(m+nlogn)) 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 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 O(mn+n2logn) time. Structural Information and Communication Complexity Structural Information and Communication Complexity Look Inside Other actions Reprints and Permissions Export citation About this Book Add to Papers Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn
2009
978-3-642-11475-5
Stability of Networks in Stretchable Graphs / Bilò, Davide; Gatto, Michael; Gualà, Luciano; Proietti, Guido; Widmayer, Peter. - 5869:(2009), pp. 100-112. (Intervento presentato al convegno 16th International Colloquium on Structural Information and Communication Complexity tenutosi a Piran, Slovenia nel May 25-27, 2009) [10.1007/978-3-642-11476-2_9].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11388/67937
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