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The quadratic minimum spanning tree problem: lower bounds via extended formulations

Renata Sotirov, Zoe Verchére

Abstract

The quadratic minimum spanning tree problem (QMSTP) is the problem of finding a spanning tree of a graph such that the total interaction cost between pairs of edges in the tree is minimized. We first show that most of the bounding approaches for the QMSTP are closely related. Then, we exploit an extended formulation for the minimum spanning tree problem to derive a sequence of relaxations for the QMSTP with increasing complexity. The resulting relaxations differ from the relaxations in the literature. Namely, our relaxations have a polynomial number of constraints and can be efficiently solved by a cutting plane algorithm. Moreover our bounds outperform most of the bounds from the literature.

The quadratic minimum spanning tree problem: lower bounds via extended formulations

Abstract

The quadratic minimum spanning tree problem (QMSTP) is the problem of finding a spanning tree of a graph such that the total interaction cost between pairs of edges in the tree is minimized. We first show that most of the bounding approaches for the QMSTP are closely related. Then, we exploit an extended formulation for the minimum spanning tree problem to derive a sequence of relaxations for the QMSTP with increasing complexity. The resulting relaxations differ from the relaxations in the literature. Namely, our relaxations have a polynomial number of constraints and can be efficiently solved by a cutting plane algorithm. Moreover our bounds outperform most of the bounds from the literature.

Paper Structure

This paper contains 15 sections, 4 theorems, 37 equations, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main results and outline
  3. The Quadratic Minimum Spanning Tree Problem
  4. An overview of lower bounds for the QMSTP
  5. The Gilmore-Lawler type bounds
  6. Assad-Xu leveling procedure
  7. Öncan-Punnen bound
  8. The RLT type bounds
  9. New lower bounds for the QMSTP
  10. Linearization-based lower bounds
  11. Strong and efficient lower bounds
  12. Numerical results
  13. Test instances
  14. Computational Results
  15. Conclusion

Key Result

Theorem 4.1

Let $G$ be a complete graph. Then, optimization problems inBasedBoundl1 and DualextendModel2 are equivalent.

Theorems & Definitions (6)

  • Remark 1
  • Theorem 4.1
  • Proposition 4.2
  • proof
  • Corollary 4.3
  • Corollary 4.4