Quadratic-Time Algorithm for the Maximum-Weight $(k, ell)$-Sparse Subgraph Problem

2025-11-25

AI Summary

This paper addresses the computational problem of finding a maximum-weight $(k,ell)$-sparse subgraph in edge-weighted graphs. To overcome a long-standing efficiency bottleneck, we propose and rigorously prove the first exact algorithm with time complexity $O(n^2 + m)$, correcting erroneous complexity analyses in prior work. Our method leverages the $(k,ell)$-sparsity matroid structure and integrates a greedy strategy with efficient dynamic data structures to enable rapid verification of sparsity constraints and incremental maintenance of candidate subgraphs. The algorithm significantly accelerates core problems in rigidity theory—including redundant rigidity testing and global rigidity subgraph identification—and successfully supports applications such as enumeration of crossing-free minimally rigid frameworks and joint identification in mechanisms. An open-source implementation is publicly available, providing a new practical tool for combinatorial optimization and structural rigidity analysis.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The family of $(k, ell)$-sparse graphs, introduced by Lorea, plays a central role in combinatorial optimization and has a wide range of applications, particularly in rigidity theory. A key algorithmic challenge is to compute a maximum-weight $(k, ell)$-sparse subgraph of a given edge-weighted graph. Although prior approaches have long provided an $O(nm)$-time solution, a previously proposed $O(n^2 + m)$ method was based on an incorrect analysis, leaving open whether this bound is achievable. We answer this question affirmatively by presenting the first $O(n^2 + m)$-time algorithm for computing a maximum-weight $(k, ell)$-sparse subgraph, which combines an efficient data structure with a refined analysis. This quadratic-time algorithm enables faster solutions to key problems in rigidity theory, including computing minimum-weight redundantly rigid and globally rigid subgraphs. Further applications include enumerating non-crossing minimally rigid frameworks and recognizing kinematic joints. Our implementation of the proposed algorithm is publicly available online.

Problem

• Develops quadratic-time algorithm for maximum-weight sparse subgraphs
• Addresses open complexity question in combinatorial optimization
• Enables faster solutions for rigidity theory problems

Innovation

• Quadratic-time algorithm for maximum-weight sparse subgraphs
• Combines efficient data structure with refined analysis
• Enables faster solutions in rigidity theory problems