A min-max regret approach for the Steiner Tree Problem with Interval Costs

TL;DR

This paper addresses the robust Min-max Regret Steiner Tree Problem with interval edge costs, proposing an ILP model, exact algorithm, and heuristics, with computational experiments demonstrating their effectiveness.

Contribution

It introduces a comprehensive approach including models and algorithms for the robust MM-RSTP with interval costs, advancing solution methods for this complex variant.

Findings

The ILP formulation effectively models the problem.

The exact algorithm solves small to medium instances efficiently.

Heuristics provide good solutions with reduced computational time.

Abstract

Let G=(V,E) be a connected graph, where V and E represent, respectively, the node-set and the edge-set. Besides, let Q \subseteq V be a set of terminal nodes, and r \in Q be the root node of the graph. Given a weight c_{ij} \in \mathbb{N} associated to each edge (i,j) \in E, the Steiner Tree Problem in graphs (STP) consists in finding a minimum-weight subgraph of G that spans all nodes in Q. In this paper, we consider the Min-max Regret Steiner Tree Problem with Interval Costs (MMR-STP), a robust variant of STP. In this variant, the weight of the edges are not known in advance, but are assumed to vary in the interval [l_{ij}, u_{ij}]. We develop an ILP formulation, an exact algorithm, and three heuristics for this problem. Computational experiments, performed on generalizations of the classical STP instances, evaluate the efficiency and the limits of the proposed methods.