Dynamic programming for spanning tree problems: application to the multi-objective case

Abstract

The aim of this paper is to provide a dynamic programming formulation for the spanning tree problem (\(\mathcal {STP}\)), which allows several instances of the classical \(\mathcal {STP}\) to be addressed. The spanning tree structure is modelled with states and transition between states, defining a state-space. Several properties are shown and optimality conditions are given. Once the theoretical fundamentals of the proposed formulation are derived, the multi-objective spanning tree problem (\(\mathcal {MOSTP}\)) is addressed. This problem arises in the telecommunications and transportation sectors. In these contexts, handling different criteria simultaneously plays a crucial role. The scientific literature provides several works that focus on the bi-objective version of the considered problem, in which only two criteria are taken into account. To the best of our knowledge, no works provide optimal methods to address the \(\mathcal {MOSTP}\) with an arbitrary number \(l\) of objective functions. In this paper we extend the proposed dynamic programming formulation to model and solve the \(\mathcal {MOSTP}\) with \(l \ge 3\) criteria.