Efficient Algorithms and Implementations for Extracting Maximum-Size $(k,ell)$-Sparse Subgraphs

2025-11-20

AI Summary

This paper addresses the problem of efficiently extracting maximum-size $(k,ell)$-sparse subgraphs from multi-graphs—a task with fundamental applications in rigidity theory and combinatorial optimization. We propose an enhanced augmenting-path algorithm that integrates three key innovations: quasi-forest initialization, dynamic edge/vertex ordering, and a two-stage optimization framework—ensuring optimality while substantially reducing the cost of augmenting operations. For the special case where the number of vertices is at least three and the graph satisfies $(k,2k)$-sparsity, we design an asymptotically faster dedicated algorithm. Experimental evaluation on both synthetic and real-world graphs demonstrates speedups of multiple orders of magnitude over state-of-the-art tools. Our implementation is open-sourced and recommended for integration into the LEMON graph algorithms library.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

A multigraph $G = (V, E)$ is $(k, ell)$-sparse if every subset $X subseteq V$ induces at most $max{k|X| - ell, 0}$ edges. Finding a maximum-size $(k, ell)$-sparse subgraph is a classical problem in rigidity theory and combinatorial optimization, with known polynomial-time algorithms. This paper presents a highly efficient and flexible implementation of an augmenting path method, enhanced with a range of powerful practical heuristics that significantly reduce running time while preserving optimality. These heuristics $unicode{x2013}$ including edge-ordering, node-ordering, two-phase strategies, and pseudoforest-based initialization $unicode{x2013}$ steer the algorithm toward accepting more edges early in the execution and avoiding costly augmentations. A comprehensive experimental evaluation on both synthetic and real-world graphs demonstrates that our implementation outperforms existing tools by several orders of magnitude. We also propose an asymptotically faster algorithm for extracting an inclusion-wise maximal $(k,2k)$-sparse subgraph with the sparsity condition required only for node sets of size at least three, which is particularly relevant to 3D rigidity when $k = 3$. We provide a carefully engineered implementation, which is publicly available online and is proposed for inclusion in the LEMON graph library.

Problem

• Develops efficient algorithms for maximum-size sparse subgraph extraction problems
• Enhances augmenting path method with heuristics to reduce computational time
• Addresses 3D rigidity applications through optimized (k,2k)-sparse subgraph algorithms

Innovation

• Augmenting path method with practical heuristics
• Edge and node ordering strategies to reduce runtime
• Asymptotically faster algorithm for maximal sparse subgraphs