Practical and Performant Enhancements for Maximization of Algebraic Connectivity

TL;DR

This paper improves a graph sparsification algorithm called MAC by developing faster solvers, better optimization strategies, and automatic edge-preservation schemes, making it more scalable and suitable for real-time applications.

Contribution

The paper introduces a specialized solver, enhanced step size strategies, and automatic connectivity schemes to advance MAC's scalability and practicality.

Findings

2x average runtime speedup with the new solver

Enhanced convergence speed and solution quality

Automatic schemes ensure graph connectivity without manual input

Abstract

Long-term state estimation over graphs remains challenging as current graph estimation methods scale poorly on large, long-term graphs. To address this, our work advances a current state-of-the-art graph sparsification algorithm, maximizing algebraic connectivity (MAC). MAC is a sparsification method that preserves estimation performance by maximizing the algebraic connectivity, a spectral graph property that is directly connected to the estimation error. Unfortunately, MAC remains computationally prohibitive for online use and requires users to manually pre-specify a connectivity-preserving edge set. Our contributions close these gaps along three complementary fronts: we develop a specialized solver for algebraic connectivity that yields an average 2x runtime speedup; we investigate advanced step size strategies for MAC's optimization procedure to enhance both convergence speed and solution quality; and we propose automatic schemes that guarantee graph connectivity without requiring manual specification of edges. Together, these contributions make MAC more scalable, reliable, and suitable for real-time estimation applications.