MIP model-based heuristics for the minimum weighted tree reconstruction problem

Abstract

We consider the minimum weighted tree reconstruction (MWTR) problem and two matheuristic methods to obtain optimal or near-optimal solutions: the Feasibility Pump heuristic and the Local Branching heuristic. These matheuristics are based on a Mixed Integer Programming model used to find feasible solutions. We discuss the applicability and effectiveness of the matheuristics to obtain solutions to the MWTR problem. The purpose of the MWTR problem is to find a minimum weighted tree connecting a set of leaves in such a way that the length of the path between each pair of leaves is greater than or equal to a given distance between the considered pair of leaves. The Feasibility Pump matheuristic starts with the Linear Programming solution, iteratively fixes the values of some variables and solves the corresponding problem until a feasible solution is achieved. The Local Branching matheuristic, in its turn, improves a feasible solution by using a local search. Computational results using two different sets of instances, one from the phylogenetic area and another from the telecommunications area, show that these matheuristics are quite effective in finding feasible solutions and present small gap values. Each matheuristic can be used independently; however, the best results are obtained when used together. For instances of the problem having up to 17 leaves, the feasible solution obtained by the Feasibility Pump heuristic is improved by the Local Branching heuristic. Noticeably, when comparing with existing based models processes that solve instances having up to 15 leaves, this achievement of the matheuristic increases the size of solved instances.