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Spanning and Splitting: Integer Semidefinite Programming for the Quadratic Minimum Spanning Tree Problem

Frank de Meijer, Melanie Siebenhofer, Renata Sotirov, Angelika Wiegele

TL;DR

This paper tackles the Quadratic Minimum Spanning Tree Problem ($QMSTP$) by developing a compact MISDP formulation that leverages graph algebraic connectivity, leading to a doubly nonnegative (DNN) relaxation augmented with CG and RLT-type cuts. A tailored Peaceman–Rachford splitting method (PRSM) is designed to efficiently solve large-scale DNN relaxations and to iteratively incorporate cuts, yielding strong bounds with practical computation times. The authors provide a thorough computational study across diverse benchmark sets, showing that their DNN bounds with cuts outperform existing SDP and RLT-based bounds, particularly for larger graphs. The results highlight semidefinite programming as a powerful tool for obtaining high-quality bounds in quadratic combinatorial problems and motivate future integration of these bounds into branch-and-bound algorithms and further strengthening via additional cuts.

Abstract

In the quadratic minimum spanning tree problem (QMSTP) one wants to find the minimizer of a quadratic function over all possible spanning trees of a graph. We present a formulation of the QMSTP as a mixed-integer semidefinite program exploiting the algebraic connectivity of a graph. Based on this formulation, we derive a doubly nonnegative relaxation for the QMSTP and investigate classes of valid inequalities to strengthen the relaxation using the Chvátal-Gomory procedure for mixed-integer conic programming. Solving the resulting relaxations is out of reach for off-the-shelf software. We therefore develop and implement a version of the Peaceman-Rachford splitting method that allows to compute the new bounds for graphs from the literature. The computational results demonstrate that our bounds significantly improve over existing bounds from the literature in both quality and computation time, in particular for graphs with more than 30 vertices. This work is further evidence that semidefinite programming is a valuable tool to obtain high-quality bounds for problems in combinatorial optimization, in particular for those that can be modelled as a quadratic problem.

Spanning and Splitting: Integer Semidefinite Programming for the Quadratic Minimum Spanning Tree Problem

TL;DR

This paper tackles the Quadratic Minimum Spanning Tree Problem () by developing a compact MISDP formulation that leverages graph algebraic connectivity, leading to a doubly nonnegative (DNN) relaxation augmented with CG and RLT-type cuts. A tailored Peaceman–Rachford splitting method (PRSM) is designed to efficiently solve large-scale DNN relaxations and to iteratively incorporate cuts, yielding strong bounds with practical computation times. The authors provide a thorough computational study across diverse benchmark sets, showing that their DNN bounds with cuts outperform existing SDP and RLT-based bounds, particularly for larger graphs. The results highlight semidefinite programming as a powerful tool for obtaining high-quality bounds in quadratic combinatorial problems and motivate future integration of these bounds into branch-and-bound algorithms and further strengthening via additional cuts.

Abstract

In the quadratic minimum spanning tree problem (QMSTP) one wants to find the minimizer of a quadratic function over all possible spanning trees of a graph. We present a formulation of the QMSTP as a mixed-integer semidefinite program exploiting the algebraic connectivity of a graph. Based on this formulation, we derive a doubly nonnegative relaxation for the QMSTP and investigate classes of valid inequalities to strengthen the relaxation using the Chvátal-Gomory procedure for mixed-integer conic programming. Solving the resulting relaxations is out of reach for off-the-shelf software. We therefore develop and implement a version of the Peaceman-Rachford splitting method that allows to compute the new bounds for graphs from the literature. The computational results demonstrate that our bounds significantly improve over existing bounds from the literature in both quality and computation time, in particular for graphs with more than 30 vertices. This work is further evidence that semidefinite programming is a valuable tool to obtain high-quality bounds for problems in combinatorial optimization, in particular for those that can be modelled as a quadratic problem.
Paper Structure (18 sections, 9 theorems, 51 equations, 1 figure, 1 table, 2 algorithms)

This paper contains 18 sections, 9 theorems, 51 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Quadratic Minimum Spanning Tree Problem
  3. MISDP formulation for the QMSTP
  4. Compact MISDP formulation for the QMSTP
  5. Valid inequalities
  6. DNN relaxation
  7. Peaceman-Rachford splitting method for the QMSTP
  8. PRSM for solving the DNN relaxation
  9. PRSM for solving the strengthened DNN relaxation
  10. Stopping criteria and post-processing of the PRSM algorithm
  11. Stopping criteria
  12. Valid lower bound
  13. Computational results
  14. Parameter setting
  15. Benchmark instances
  16. ...and 3 more sections

Key Result

Proposition 1

Let $G$ be a simple graph on $n \geq 3$ vertices and $L$ be the Laplacian matrix of $G$. Moreover, let $\widetilde{\beta} \in \mathbb{R}$, $\widetilde{\beta} \geq 0$, and $\widetilde{\alpha} =\frac{\widetilde{\beta}}{n}$. Then $a(G) \geq \widetilde{\beta}$ if and only if $L + \widetilde{\alpha} \ma

Figures (1)

  • Figure 1: $G = K_4$ with given edge and vertex labeling.

Theorems & Definitions (12)

  • Proposition 1
  • proof
  • Proposition 2
  • Theorem 1: de2023misdp
  • Corollary 1
  • Proposition 3
  • Proposition 4
  • Theorem 2: Theorem 3.15 in Hu2023
  • Theorem 3
  • proof
  • ...and 2 more