Spectral Graph Theoretic Methods for Enhancing Network Robustness in Robot Localization

TL;DR

This paper introduces spectral graph theoretic methods, including Cheeger cuts and MILP, to optimize network robustness for improved robot localization, demonstrating significant computational efficiency improvements.

Contribution

It develops novel Cheeger cut techniques and MILP formulations to efficiently solve algebraic connectivity maximization problems in robot localization networks.

Findings

Enhanced run times on synthetic datasets

Effective Cheeger cuts for network robustness

High-quality solutions from greedy heuristics

Abstract

This paper addresses the optimization of edge-weighted networks by maximizing algebraic connectivity to enhance network robustness. Motivated by the need for precise robot position estimation in cooperative localization and pose-graph sparsification in Simultaneous Localization and Mapping (SLAM), the algebraic connectivity maximization problem is formulated as a Mixed Integer Semi-Definite Program (MISDP), which is NP-hard. Leveraging spectral graph theoretic methods, specifically Cheeger's inequality, this work introduces novel "Cheeger cuts" to strengthen and efficiently solve medium-scale MISDPs. Further, a new Mixed Integer Linear Program (MILP) is developed for efficiently computing Cheeger cuts, implemented within an outer-approximation algorithm for solving the MISDP. A greedy k-opt heuristic is also presented, producing high-quality solutions that serve as valid lower bounds for Cheeger cuts. Comprehensive numerical analyses demonstrate the efficacy of strengthened cuts via substantial improvements in run times on synthetic and realistic robot localization datasets.