Mathematical Programming

, Volume 128, Issue 1–2, pp 123–148

# Modeling hop-constrained and diameter-constrained minimum spanning tree problems as Steiner tree problems over layered graphs

• Luis Gouveia
• Luidi Simonetti
• Eduardo Uchoa
Full Length Paper Series A

## Abstract

The hop-constrained minimum spanning tree problem (HMSTP) is an NP-hard problem arising in the design of centralized telecommunication networks with quality of service constraints. We show that the HMSTP is equivalent to a Steiner tree problem (STP) in an appropriate layered graph. We prove that the directed cut model for the STP defined in the layered graph, dominates the best previously known models for the HMSTP. We also show that the Steiner directed cuts in the extended layered graph space can be viewed as being a stronger version of some previously known HMSTP cuts in the original design space. Moreover, we show that these strengthened cuts can be combined and projected into new families of cuts, including facet defining ones, in the original design space. We also adapt the proposed approach to the diameter-constrained minimum spanning tree problem (DMSTP). Computational results with a branch-and-cut algorithm show that the proposed method is significantly better than previously known methods on both problems.

## Keywords

Networks/graphs: tree algorithms Integer programming: formulations Cutting planes

## Mathematics Subject Classification (2000)

90C11 Mixed integer programming 90C27 Combinatorial optimization 90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut

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