Abstract
This article introduces a new algorithm for computing the set of supported non-dominated points in the objective space and all the corresponding efficient solutions in the decision space for the multi-objective spanning tree (MOST) problem. This algorithm is based on the connectedness property of the set of efficient supported solutions and uses a decomposition of the weight set in the weighting space defined for a parametric version of the MOST problem. This decomposition is performed through a space reduction approach until an indifference region for each supported non-dominated point is obtained. An adjacency relation defined in the decision space is used to compute all the supported efficient spanning trees associated to the same non-dominated supported point as well as to define the indifference region of the next points. An in-depth computational analysis of this approach for different types of networks with three objectives is also presented.
Highlights
The minimum spanning tree (MST) problem is a combinatorial optimization problem with many real-world applications [2, 37, 38], which are present in almost our daily life aspects
The number of separating hyperplanes is at most mðm À 1Þ=2, as each edge can only be swapped with the other m À 1 edges, and amongst each pair of edges, independently of which edge is going in/out of the tree, only one separated hyperplane is generated
The large difference between the number of regions obtained by our algorithm and the maximum number of possible regions suggests that an algorithm that computes firstly the set of indifference regions might generate a large amount of regions leading to the same solution
Summary
The minimum spanning tree (MST) problem is a combinatorial optimization problem with many real-world applications [2, 37, 38], which are present in almost our daily life aspects The algorithm starts by calculating a lexicographical optimal spanning tree w.r.t. one objective function, and by searching amongst its adjacent trees, it calculates the weightings for which that tree is optimal for the scalarized version of MOST with respect to the weighted sum (so called parametric problem). To the bi-objective version described in [2], we calculate the weight set region for which every supported solution is optimal for the parametric problem. Such a region in the general MOST problem is a polytope, which we refer to as an indifference region. This section introduces the minimum spanning tree problem along with some fundamental results, the presentation of its multi-objective variant, the weightedsum spanning tree problem, and some algorithmic aspects for the three-objective spanning tree problem
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
