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.

翻译：在图上进行长期状态估计仍具挑战性，因为现有图估计算法在大型长期图上的扩展性较差。为解决此问题，本研究改进了当前最先进的图稀疏化算法——代数连通度最大化（MAC）。MAC是一种通过最大化代数连通度来保持估计性能的稀疏化方法，代数连通度作为谱图性质与估计误差直接相关。然而，MAC的计算复杂度仍使其难以在线应用，且需用户手动预定义保持连通性的边集。本研究的贡献从三个互补层面弥补了这些不足：我们开发了针对代数连通度的专用求解器，实现了平均2倍的运行加速；研究了MAC优化过程中先进的步长策略，以提升收敛速度和解的质量；提出了自动保证图连通性的方案，无需人工指定边集。这些贡献共同使MAC更具可扩展性、可靠性，并适用于实时估计应用。