The k k -Cardinality Tree Problem: Reformulations and Lagrangian Relaxation
Given an undirected defining graph for the k k -Cardinality Tree Problem (KCTP), an associated directed graph involving two additional vertices is introduced in this paper and gives rise to two compact reformulations of the problem. For the first one, connectivity of feasible solutions is enforced through multicommodity flows while, for the other, lifted MillerTuckerZemlin constraints are used. Comparing the two reformulations, much stronger Linear Programming relaxation bounds are obtained from the first one, albeit at much higher CPU times. However, a Branch-and-Bound algorithm based on the second reformulation proved much more effective and managed to obtain, for the first time, optimality certificates for a large number of KCTP instances from the literature. Additionally, for some instances where optimality could not be proven within the given pre-specified CPU time limit, new best upper bounds were generated. Finally, a Lagrangian heuristic based on the first reformulation was also implemented and proved capable of generating feasible KCTP solutions comparable in quality with the best overall results obtained by metaheuristic based heuristics found in the literature. For our test cases, Lagrangian upper bounds are no more than 3.8% away from the best upper bounds known. Additionally, several new best upper bounds and optimality certificates were obtained by the heuristic. Corresponding Lagrangian heuristic CPU times, however, are typically higher than those associated with their competitors.
lagrangian relaxation k k -cardinality tree problem k k k k k integer programming