Asymptotically faster algorithms for recognizing $(k,\ell)$-sparse graphs

2026-04-14

AI Summary

This work addresses the efficient recognition of $(k,\ell)$-sparse graphs: given a graph, it determines whether the graph satisfies $(k,\ell)$-sparsity and, if not, explicitly outputs a vertex set violating the sparsity condition. We propose a novel algorithm that integrates bounded-indegree orientation, reduction to rooted arc-connectivity, augmenting-path techniques, and a divide-and-conquer strategy based on centroid decomposition. For the first time, this approach achieves subquadratic and even near-linear time complexity in classical parameter regimes: specifically, $O(n\sqrt{n})$ when $0 \leq \ell \leq k$, $O(n\sqrt{n\log n})$ when $k < \ell < 2k$, and $O(n^2)$ or $O(n^2\log n)$ when $2k \leq \ell < 3k$, while also producing an explicit certificate of sparsity violation.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• 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 problem is to decide whether a given graph is $(k,\ell)$-sparse and, if not, to produce a vertex set certifying the failure of sparsity. While pebble game algorithms have long yielded $O(n^2)$-time recognition throughout the classical range $0 \leq \ell < 2k$, and $O(n^3)$-time algorithms in the extended range $2k \leq \ell < 3k$, substantially faster bounds were previously known only in a few special cases. We present new recognition algorithms for the parameter ranges $0 \le \ell \le k$, $k < \ell < 2k$, and $2k \leq \ell < 3k$. Our approach combines bounded-indegree orientations, reductions to rooted arc-connectivity, augmenting-path techniques, and a divide-and-conquer method based on centroid decomposition. This yields the first subquadratic, and in fact near-linear-time, recognition algorithms throughout the classical range when instantiated with the fastest currently available subroutines. Under purely combinatorial implementations, the running times become $O(n\sqrt n)$ for $0 \leq \ell \leq k$ and $O(n\sqrt{n\log n})$ for $k< \ell <2k$. For $2k \leq \ell < 3k$, we obtain an $O(n^2)$-time algorithm when $\ell \leq 2k+1$ and an $O(n^2\log n)$-time algorithm otherwise. In each case, the algorithm can also return an explicit violating set certifying that the input graph is not $(k,\ell)$-sparse.

Problem

• (k, ell)-sparse graphs
• graph recognition
• sparsity certification
• combinatorial optimization
• rigidity theory

Innovation

• (k,ℓ)-sparse graphs
• subquadratic algorithms
• centroid decomposition
• bounded-indegree orientations
• rigidity theory