Algorithms to solve the knapsack constrained maximum spanning tree problem

Abstract

The knapsack problem and the minimum spanning tree problem are both fundamental in operations research and computer science. We are concerned with a combination of these two problems. That is, we are given a knapsack of a fixed capacity, as well as an undirected graph where each edge is associated with profit and weight. The problem is to fill the knapsack with a feasible spanning tree such that the tree profit is maximized. We prove this problem NP-hard, present upper and lower bounds, develop a branch-and-bound algorithm to solve the problem to optimality and propose a shooting method to accelerate computation. We evaluate the developed algorithm through a series of numerical experiments for various types of test problems.

A Lagrangian Decomposition/Evolutionary Algorithm Hybrid for the Knapsack Constrained Maximum Spanning Tree Problem

“…The later articles from Xue [6] (weight-constrained MST ) and Jüttner [7] (constrained minimum cost spanning tree problem) deal with two similar primal-dual algorithms. Recently, Yamada et al [1] (KCMST problem) also described a LR approach, which yields feasible heuristic solutions, too. These are further improved by a 2-opt local search.…”

“…In comparison, in the LR approach from [1,7] the knapsack constraint is relaxed and only the MST problem remains. This approach therefore fulfills the integrality property and, thus, is in general weaker than our LD.…”

“…As in [1], we consider instances based on complete graphs K |V |,γ and planar graphs P |V |,|E|,γ . Parameter γ represents the type of correlation between profits and weights: uncorrelated ("u"): p e and w e , e ∈ E, are independently chosen from the integer interval [1, 100]; weakly correlated ("w"): w e is chosen as before, and p e := 0.8w e + v e , where v e is randomly selected from [1,20]; strongly correlated ("s"): w e is chosen as before, and p e := 0.9w e + 10 .…”

“…For details on the methods used to construct the planar graphs, we refer to [1,8]. Since we could not obtain the original instances, we created them in the same way by our own.…”

“…Yamada et al [1] gave a proof for the KCMST problem's N P-hardness. An exemplary instance and its solution are shown in Fig.…”