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Optimal Robust Network Design: Formulations and Algorithms for Maximizing Algebraic Connectivity

Neelkamal Somisetty, Harsha Nagarajan, Swaroop Darbha

TL;DR

This article introduces a novel upper bounding algorithm based on the principal minor characterization of positive semidefinite matrices and discusses a degree-constrained lower bounding formulation inspired by robust network structures, and proposes a maximum cost heuristic with low computational complexity to identify high-quality feasible solutions for instances involving up to 100 nodes.

Abstract

This paper focuses on designing edge-weighted networks, whose robustness is characterized by maximizing algebraic connectivity, or the second smallest eigenvalue of the Laplacian matrix. This problem is motivated by cooperative vehicle localization, where accurately estimating relative position measurements and establishing communication links are essential. We also examine an associated problem where every robot is limited by payload, budget, and communication to pick no more than a specified number of relative position measurements. The basic underlying formulation for these problems is nonlinear and is known to be NP-hard. Our approach formulates this problem as a Mixed Integer Semi-Definite Program (MISDP), later reformulated into a Mixed Integer Linear Program (MILP) for obtaining optimal solutions using cutting plane algorithms. We introduce a novel upper-bounding algorithm based on principal minor characterization of positive semi-definite matrices and discuss a degree-constrained lower bounding formulation inspired by robust network structures. In addition, we propose a maximum cost heuristic with low computational complexity to identify high-quality feasible solutions for instances involving up to one hundred nodes. We show extensive computational results corroborating our proposed methods.

Optimal Robust Network Design: Formulations and Algorithms for Maximizing Algebraic Connectivity

TL;DR

This article introduces a novel upper bounding algorithm based on the principal minor characterization of positive semidefinite matrices and discusses a degree-constrained lower bounding formulation inspired by robust network structures, and proposes a maximum cost heuristic with low computational complexity to identify high-quality feasible solutions for instances involving up to 100 nodes.

Abstract

This paper focuses on designing edge-weighted networks, whose robustness is characterized by maximizing algebraic connectivity, or the second smallest eigenvalue of the Laplacian matrix. This problem is motivated by cooperative vehicle localization, where accurately estimating relative position measurements and establishing communication links are essential. We also examine an associated problem where every robot is limited by payload, budget, and communication to pick no more than a specified number of relative position measurements. The basic underlying formulation for these problems is nonlinear and is known to be NP-hard. Our approach formulates this problem as a Mixed Integer Semi-Definite Program (MISDP), later reformulated into a Mixed Integer Linear Program (MILP) for obtaining optimal solutions using cutting plane algorithms. We introduce a novel upper-bounding algorithm based on principal minor characterization of positive semi-definite matrices and discuss a degree-constrained lower bounding formulation inspired by robust network structures. In addition, we propose a maximum cost heuristic with low computational complexity to identify high-quality feasible solutions for instances involving up to one hundred nodes. We show extensive computational results corroborating our proposed methods.
Paper Structure (27 sections, 3 theorems, 24 equations, 6 figures, 5 tables, 1 algorithm)

This paper contains 27 sections, 3 theorems, 24 equations, 6 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Maximization of Algebraic connectivity
  3. Cooperative vehicle localization with noisy measurements
  4. An associated problem of interest
  5. Mathematical Formulations to Maximize Algebraic connectivity
  6. Mixed integer semi-definite program
  7. Relaxation & eigenvector cuts
  8. Upper Bounds on Optimal Algebraic connectivity of Networks
  9. Principal minor characterization of PSD matrices
  10. Principal minor-based relaxation formulations
  11. Degree-constrained Lower Bounding Formulation
  12. Maximum Cost Heuristic
  13. Maximum cost heuristic for the associated problem
  14. Computational Results
  15. Instance generation
  16. ...and 12 more sections

Key Result

Lemma 1

Let $(\gamma^*, \mathbf{x}^*)$ and $(\gamma_2^*, \mathbf{x}_2^*)$ be the optimal solutions of $\mathcal{F}_1$ and $\mathcal{F}_2$, respectively. Then, $\gamma^* = \gamma_2^*$, and the associated feasible solutions, $\mathbf{x}^* = \mathbf{x}_2^*$.

Figures (6)

  • Figure 1: Degree histogram of optimal/best-known solutions for instances with 8, 10, 12 and 15 nodes, averaged over fifty instances of each size. The y-axis denotes the percentage of nodes with a specific degree in an instance.
  • Figure 2: The best known feasible networks for varying instance sizes obtained using DCLBF ${\mathcal{F}}_{k}^{l}$ in \ref{['eq:F_d_k']} with various $k$ values. Edge weights are omitted for clarity.
  • Figure 3: Comparing the quality of the solutions and run times for different values of $k$ for ten node instances.
  • Figure 4: Comparing the solutions and runtimes of the proposed maximum cost heuristic (MCH) with parameters ($h_1, h_2$) set to (5, 3) against FSM son2010building for networks up to one hundred nodes. The node degree upper bound is set to five.
  • Figure 5: Comparing the solutions and runtimes of the maximum cost heuristic (MCH) for 25-node instances ($k=10$) with varying heuristic parameters. Line types represent variations in $h_1$, while colors denote changes in $h_2$.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Lemma 1
  • proof
  • Definition 1
  • Definition 2
  • Proposition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 2
  • Remark 4
  • ...and 1 more