A Branch and Bound Algorithm for the Robust Spanning Tree Problem with Interval Data
A branch and bound algorithm for the robust spanning tree problem with interval data
R. Montemanni∗, L.M. Gambardella
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.
1
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
R. Montemanni∗, L.M. Gambardella
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.
1
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
References: [1] I. Aron and P. Van Hentenryck. A constraints satisfaction approach to the robust spanning tree problem with interval data. In preparation. Computer Science Department, Brown University, May 2002. [2] I. Aron and P. Van Hentenryck. On the complexity of the robust spanning tree with interval data. Operations Research Letters, to appear. [3] D.P. Bertsekas and R. Gallagher. Data Networks. Prentice-Hall, Englewood Cliffs, NJ, 1987. [4] A. Cayley. A theorem on trees. Quarterly Journal of Pure and Applied Mathematics, 23:376–378, 1889. [5] J.J. Dongarra. Performance of various computers using standard linear algebra software in a fortran environment. Technical Report CS-89-85, University of Tennessee, July 2003. [6] H.N. Gabow. Two algorithms for generating weighted spanning trees in order. SIAM Journal on Computing, 6(1):139–150, March 1977. 10 [7] P. Kouvelis and G. Yu. Robust Discrete Optimization and its applications. Kluwer Academic Publishers, 1997. [8] G.L. Kozina and V.A. Perepelista. Interval spanning trees problem: solvability and computational complexity. Interval Computations, 1:42–50, 1994. [9] J.B. Kruskal. On the shortest spanning subtree of a graph and the travelling salesman problem. Preceedings of the American Mathematical Society, 7:48–50, 1956. [10] R. Montemanni, L.M. Gambardella, and A.V. Donati. A branch and bound algorithm for the robust shortest path problem with interval data. Operations Research Letters, to appear. [11] R.C. Prim. Shortest connection networks and some generalizations. Bell System Technical Journal, 36:1389–1401, 1957. [12] H. Yaman, O.E. Kara¸an, and M.C. Pinar. The robust spanning tree problem with s ¸ interval data. Operations Research Letters, 29:31–40, 2001. 11