Spectral Graph Theoretic Methods for Enhancing Network Robustness in Robot Localization

TL;DR

This work addresses robust network design for robot localization by maximizing the algebraic connectivity $\\lambda_2(\\mathbf{L(x)})$ of an edge-weighted graph under a budget $q$ edges, an NP-hard MISDP problem. It introduces Cheeger constant-based valid cuts that exploit Cheeger\\'s inequality $c_f \\lambda_2(G) \\le \\phi(G) \\le \\sqrt{2\\lambda_2(G)}$ to tighten bounds, and a MILP formulation for computing the Cheeger constant that outperforms a MIQCP. It also develops a greedy $k$-opt heuristic to rapidly sparsify pose-graph SLAM graphs and provide strong lower bounds. Computational results show substantial runtime improvements, enabling optimal or near-optimal solutions on medium-scale networks and high-quality sparsifications for large real-world datasets, with direct impact on cooperative localization and SLAM.

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.

Figures (4)

• Figure 1: Convex hull as the feasible region (in red) for every line $\{i,j\} \in E$.
• Figure 2: Comparative analysis of heuristics (3-opt, 2-opt, and 1-opt) with $m=20$ for spanning tree solutions across different network sizes ($n$). The gap and run times are averaged over fifty random instances for each $n$.
• Figure 3: Comparison of the greedy 1-opt and 2-opt heuristics with parameter $m = 20$ against the MAC algorithm doherty2022spectral for different graph sizes. The left graph displays the algebraic connectivity, while the right graph illustrates the run times.
• Figure 4: Pose-graph optimization for the CSAIL dataset. The sequence shows: (a) the original graph with all measurements; (b) 20% edge sparsification using greedy 1-opt heuristic to maximize algebraic connectivity with $m=30$; (c) 20% edge sparsification using the MAC algorithm doherty2022spectral; and (d) 20% edge sparsification through random edge selection. In (b), (c), and (d), the sparsified networks (red) are superimposed on the original graph (blue), facilitating a visual comparison of the sparsification effects.

Theorems & Definitions (6)

• Definition 1
• Definition 2
• Remark 1
• Theorem 1
• Lemma 1
• proof