Lagrangian Relax-and-Cut and Hybrid Methods for the Bounded Diameter and the Hop Constrained Minimum Spanning Tree Problems

The Bounded Diameter Minimum Spanning Tree problem (BDMST) and the Hop Constrained Minimum Spanning Tree problems (HCMST) are NP-hard combinatorial optimization problems which have their main application in network design. In this thesis an existing relax-and-cut approach for finding lower bounds and approximate solutions to those problems is enhanced and extended, and a hybrid algorithm based on the relax-and-cut approach as well as on an existing metaheuristic, namely an ant colony optimization (ACO), is presented. The enhanced relax-and-cut (R&C) approach is based on an integer linear programming (ILP) formulation which relies on so called jump constraints. The number of jump constraints in this formulation is exponential by means of the instance size. Therefore, violated constraints are identified and relaxed on the fly. The enhanced R&C algorithm is a so called non deleayed relax-and-cut algorithm which is based on subgradient optimization. Since the number of separated jump inequalities can be large, a sophisicated management of a pool of such constraints is used. The two main extensions to this R&C approach are the initial identification of jump constraints with corresponding dual variables and a generalization of jump constraints. The metaheuristic utilized for the hybrid algorithm is an ant colony optimization (ACO) algorithm. ACO algorithms exploit the ability of ants finding short paths between their nest and food sources by depositing pheromone. This works as a positive feedback system. The concept of the hybrid algorithm is the utilization of information obtained by the relax-and-cut approach as a heuristic component in the ACO algorithm that is mixed with the pheromone information of the ACO algorithm. Computational experiments have been performed on previously published benchmark instances. The results have shown that most of the enhancements to the R&C algorithm have lead to significant improvements especially for the lower bounds compared to the original R&C algorithm.