# Solving Steiner Tree Problems in Graphs with Lagrangian Relaxation

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## Abstract

This paper presents an algorithm to obtain near optimal solutions for the Steiner tree problem in graphs. It is based on a Lagrangian relaxation of a multi-commodity flow formulation of the problem. An extension of the subgradient algorithm, the *volume algorithm,* has been used to obtain lower bounds and to estimate primal solutions. It was possible to solve several difficult instances from the literature to proven optimality without branching. Computational results are reported for problems drawn from the *SteinLib* library.

Steiner trees Lagrangian relaxation

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