Computational comparisons of different formulations for the Bilevel Minimum Spanning Tree Problem

International audienceLet be given a graph G = (V, E) whose edge set is partitioned into a set R of red edges and a set B of blue edges, and assume that red edges are weighted and contain a spanning tree of G. Then, the Bilevel Minimum Spanning Tree Problem (BMSTP) is that of pricing (i.e., weighting) the blue edges in such a way that the total weight of the blue edges selected in a minimum spanning tree of the resulting graph is maximized. In this paper we present different mathematical formulations for the BMSTP based on the properties of the Minimum Spanning Tree Problem and the bilevel optimization. We establish a theoretical and empirical comparison between these new formulations and we also provide reinforcements that together with a proper formulation are able to solve medium to big size instances random instances. We also test our models in instances already existing in the literature

A Branch-and-cut-and-price algorithm for the Stackelberg Minimum Spanning Tree Game

International audienceThe Stackelberg Minimum Spanning Tree Game (StackMST) is defined in terms of a graph G = (V, B ∪ R), with two disjoint sets of edges, blue B and red R, and costs {c e ≥ 0 : e ∈ R} defined for the red edges. Once the leader of the game defines prices {p e : e ∈ B} to the blue edges, the follower chooses a minimum weight spanning tree (V, E T), at cost e∈B∩E T p e + e∈R∩E T c e. The goal is to find prices to maximize the revenue e∈B∩E T p e collected by the leader. We introduce a reformulation and a Branch-and-cut-and-price algorithm for StackMST. The reformulation is obtained after applying KKT optimality conditions to a StackMST non-compact Bilevel Linear Programming formulation and is strengthened with a partial rank-1 RLT and with valid inequalities from the literature. We also implemented a Branch-and-cut algorithm for an extended formulation derived from another in the literature. A preliminary computational study comparing both methods is also presented