Steiner trees and Polyhedra

this paper, we study polyhedra STP(G,S) and CON(G,S). We describe a new class of facet defining inequalities for the STP(G,S) that generalizes the families of constraints so called Steiner partition inequalities and odd hole inequalities introduced by Chopra and Rao [3]. We show that these inequalities may define facets for 2-trees, which invalidates a conjecture of Chopra and Rao [4]. We also discuss the closely related Steiner connected subgraph polytope. We describe some procedures of construction of facets from facets for CON(G,S). Using this, we obtain a complete description of both CON(G,S) and STP(G,S) for a special case of series-parallel graphs. Computational applications are also discussed. 2 Valid inequalitie

A framework for routing and spectrum assignment in optical networks, driven by combinatorial properties

International audienceThe routing and spectrum assignment problem is an NP-hard problem that has received increasing attention during the last years. The majority of existing models for the problem uses edge-path formulations where variables are associated with all possible routing paths so that the number of variables grows exponentially with the size of the instance. To bypass this difficulty, precomputed subsets of all possible paths per demand are typically used, which cannot guarantee optimality of the solutions in general. Our contribution is to provide a framework for the use of edge-path formulations to minimize the spectrum width of a solution. For that, we select an appropriate subset of paths to operate on with the help of combinatorial properties in such a way that optimality of the solution can be guaranteed. Computational results indicate that our approach is indeed promising to solve the routing and spectrum assignment problem

Lagrangian decomposition, metaheuristics, and hybrid approaches for the design of the last mile in fiber optic networks

Abstract. We consider a generalization of the (Price Collecting) Steiner Tree Problem on a graph with special redundancy requirements for customer nodes. The problem occurs in the design of the last mile integer linear program and apply Lagrangian Decomposition to obtain relatively tight lower bounds as well as feasible solutions. Furthermore, a Variable Neighborhood Search and a GRASP approach are described, utilizing a new construction heuristic and special neighborhoods. In particular, hybrids of these methods are also studied and turn out to often perform superior. By comparison to previously published exact methods we show that our approaches are applicable to larger problem instances, while providing high quality solutions together with good lower bounds