Problema de Árvore Geradora Mínima com Restrição de Grau Mínima e Centrais e Terminais Fixos

Abstract

The Min-Degree Constrained Minimum Spannig Tree - MD-MST is to find a minimum spanning tree of a graph where each vertex is a leaf of the tree or satisfies a constraint of minimum degree. The leaf vertices are called terminals and the others are the central vertices. We define and study a variation of this problem, which we denote MDF-MST, where the terminal and central vertices are fixed. We show that the problem is NP-Hard and is in FPT, parameterized by the number of central vertices. We also identify cases where the problem becomes polynomial. We propose several integer programming formulations for the problem and compare the quality of lower bound generated by their linear relaxations. We propose and teste a Lagrangian Relaxation for the problem, which we also use to define Lagrangian heuristics. We define greedy heuristics, a VND Local search and a VNS heuristic. We present a Benders’s Decomposition. We propose a new general heuristic that combines ingredients from the Benders’s decomposition with subgradient method, which we call subgradient heuristic. We apply this heuristic to the MDF-MST. All these algorithms have been implemented, tested and compared among them and with the CPLEX solver. The computational efficiency of the proposed algorithms, especially the Lagrangian heuristics, is comparable with that of CPLEX, and even better in several cases. Some of these algorithms were adapted for the MD-MST and DC-MST (inthelatter,thedegreeconstraintisofmaximumdegree). Whencomparingthecomputational results with the literature, we conclude that the algorithms are competitive.