# A Branch and Bound Algorithm for the Robust Spanning Tree Problem with Interval Data

**Topics:**Minimum spanning tree, Spanning tree, Graph theory

**Pages:**16 (5768 words)

**Published:**May 15, 2013

Istituto Dalle Molle di Studi sull’Intelligenza Artiﬁciale (IDSIA) Galleria 2, CH-6928 Manno-Lugano, Switzerland

Abstract The robust spanning tree problem is a variation, motivated by telecommunications applications, of the classic minimum spanning tree problem. In the robust spanning tree problem edge costs lie in an interval instead of having a ﬁxed value. Interval numbers model uncertainty about the exact cost values. A robust spanning tree is a spanning tree whose total cost minimizes the maximum deviation from the optimal spanning tree over all realizations of the edge costs. This robustness concept is formalized in mathematical terms and is used to drive optimization. In this paper a branch and bound algorithm for the robust spanning tree problem is proposed. The method embeds the extension of some results previously presented in the literature and some new elements, such as a new lower bound and some new reduction rules, all based on the exploitation of some peculiarities of the branching strategy adopted. Computational results obtained by the algorithm are presented. The technique we propose is up to 210 faster than methods recently appeared in the literature. Keywords: Branch and bound, robust optimization, interval data, spanning tree problem.

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Introduction

This paper presents a branch and bound algorithm for a robust version of the minimum spanning tree problem where edge costs lie in an interval instead of having a ﬁxed value. Each interval is used to model uncertainty about the real value of the respective cost, which can take any value in the interval, independently from the costs associated with the other edges of the graph. Adopting the model described above, the classic optimality criterion of the minimum spanning tree problem (where a ﬁxed cost is associated with each edge of the graph) does not apply anymore, and the classic polynomial-time algorithms (Kruskal [9] and Prim [11]) cannot be used. A more complex optimization criterion has then to be adopted. We have chosen the relative robustness criterion (see Kouvelis and Yu [7]). The study has practical motivations, and in particular there are some applications in the ﬁeld of telecommunications. Consider a supervisor node in a data network where transmission lines are subject to uncertain delays, that wants to send a control message to all other nodes in the network. The supervisor node generally wants to broadcast the message over a robust spanning tree, in order to have a relatively quick broadcast whatever the situation in the network is (see Bertsekas and Gallagher [3] for a more detailed ∗ Corresponding

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description of the problem). A second application concerns the design of communication networks where routing delays on edges are uncertain, since they depend on the network traﬃc. The ideal network guarantees good performance whatever is the real traﬃc, i.e. a robust spanning tree is desirable (see Kouvelis and Yu [7] for more details). In the literature there are some other studies related to robust versions of the minimum spanning tree problem. Kozina and Perepelista [8] deﬁned an order relation on the set of feasible solutions and generated a Pareto set. Aron and Van Hentenryck [2] proved that the problem is N P-hard. In Yaman et al. [12] a mixed integer programming formulation and a preprocessing technique are presented. The present paper describes a branch and bound algorithm, based on an adaptation of the method developed in Montemanni et al. [10] for the robust shortest path problem. The algorithm incorporates an extension of the preprocessing rules described in [12] and some new concepts which signiﬁcantly contribute to the eﬃciency of the method. Another branch and bound approach to the robust spanning tree problem has been...

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