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

TL;DR

This paper develops formulations and algorithms to maximize algebraic connectivity in edge-weighted networks, enhancing robustness for applications like cooperative vehicle localization, through MISDP, MILP, and heuristic methods.

Contribution

It introduces novel MISDP and MILP formulations, a principal minor-based upper bound, and a low-complexity heuristic for robust network design.

Findings

Proposed algorithms effectively solve large instances with up to 100 nodes.

New bounds improve solution quality and computational efficiency.

Heuristic provides high-quality feasible solutions quickly.

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.