A branch and cut algorithm for minimum spanning trees under conflict constraints

Abstract

We study approaches for the exact solution of the NP-hard minimum spanning tree problem under conflict constraints. Given a graph G(V, E) and a set C ⊂ E × E of conflicting edge pairs, the problem consists of finding a conflict-free minimum spanning tree, i.e. feasible solutions are allowed to include at most one of the edges from each pair in C. The problem was introduced recently in the literature, with several results on its complexity and approximability. Some formulations and both exact and heuristic algorithms were also discussed, but computational results indicate considerably large duality gaps and a lack of optimality certificates for benchmark instances. In this paper, we build on the representation of conflict constraints using an auxiliary conflict graphĜ(E,C), where stable sets correspond to conflict-free subsets of E. We introduce a general preprocessing method and a branch and cut algorithm using an IP formulation with exponentially sized classes of valid inequalities for both the spanning tree and the stable set polytopes. Encouraging computational results indicate that the dual bounds of our approach are significantly stronger than those previously available, already in the initial LP relaxation, and we are able to provide new feasibility and optimality certificates.

A cutting-plane algorithm for the Steiner team orienteering problem

“…Instead of selecting all the violated cuts found, we only add to Γ ψ the most violated cut (if any) and the ones that are sufficiently orthogonal to it. As verified in several works (see, e.g., [37,38,39]), this strategy is able to balance the strength and diversity of the cuts separated, while limiting the model size. Here, this strategy is applied to select both GCCs and CCs, but separately (lines 7 and 8, Figure 7).…”

“…and is valid for the region Φ determined by the original knapsack constraint (37), as defined by (38). Now, let the set D ⊆ P\C identify the indexes of the y variables to be up-lifted.…”

A multiethnic genetic approach for the minimum conflict weighted spanning tree problem

“…When a conflict-free solution is not found, these heuristics return the number of conflict pairs present in the solutions. In [17] a branch-and-cut approach was proposed. This algorithm was based on the concept of conflict grapĥ G(E, C) in which each edge of the original graph G is mapped in a node ofĜ and there is an edge between two vertices inĜ if and only if the corresponding edges in G are in conflict.…”

“…Finally, very recently, another branch-and-cut algorithm for MSTC was introduced in [2]. Thanks to a new set of valid inequalities, based on combined properties belonging to any feasible solution, this last algorithm outperforms the ones proposed in [17].…”

Two dependency constrained spanning tree problems

“…Another NP‐hard variation of MST consists in imposing conflict constraints over pairs of edges (Darmann et al., ; Zhang et al., ; Samer and Urrutia, ). A conflict between a pair of edges means that at most one of them may take part in a solution.…”