The robust spanning tree problem with interval data

Abstract

Motivated by telecommunications applications we investigate the minimum spanning tree problem where edge costs are interval numbers. Since minimum spanning trees depend on the realization of the edge costs, we deÿne the robust spanning tree problem to hedge against the worst case contingency, and present a mixed integer programming formulation of the problem. We also deÿne some useful optimality concepts, and present characterizations for these entities leading to polynomial time recognition algorithms. These entities are then used to preprocess a given graph with interval data prior to the solution of the robust spanning tree problem. Computational results show that these preprocessing procedures are quite e ective in reducing the time to compute a robust spanning tree.

Minmax regret approach and optimality evaluation in combinatorial optimization problems with interval and fuzzy weights

“…Since every possibly optimal solution X is composed of possibly optimal elements and every optimal solution under a worst case configuration for X is composed of possibly optimal elements, the non-possibly optimal elements do not influence the computation of an optimal minmax regret solutions. In consequence, they can be removed from E. This general property has been proved for some particular problems in [19,27].…”

“…The deviation interval for a given element can be efficiently computed if P is a matroidal problem, for instance P is Minimum Spanning Tree. Making use of the results obtained in [21,27], it is easy to show that in this case δ f = δ(w w w − {f } ) and δ f = δ(w w w + {f } ). However, this result is not valid for all problems P.…”

“…Similarly, a solution (an element) is necessarily optimal if it is optimal for all configurations (the event that it will be optimal is sure). The notions of possible and necessary optimality of solutions and elements have been already introduced in the literature for some particular problems: in [8,9,10,12,14] for scheduling problems, in [21] for matroidal problems, in [16,17,18] for linear programming, in [19] for shortest path and in [27] for minimum spanning tree. In this paper, we generalize the optimality notions to all combinatorial optimization problems with interval weights.…”

“…Namely, in the minmax regret approach we seek a solution that minimizes the maximal regret and this approach to combinatorial optimization has been extensively studied in the recent literature (see, e.g. [2,3,5,6,19,20,24,27] and [4] for a recent survey).…”

Performance Analysis in Robust Optimization

“…The recipe that was used to derive this MIP formulation can be applied to any combinatorial optimization problem with uncertainty in the costs, and for which the nominal problem (P) can be solved by using its linear relaxation. For more information about the derivation of the robust counterpart that arises from the absolute regret concept we refer to [27].…”