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

TL;DR

This work resolves the longstanding question of whether a maximum-weight $(k, \ell)$-sparse subgraph can be found in quadratic time. It introduces a component-based pebble-game framework that maintains $(k, \ell)$-components and uses a small set of data-structural primitives to decide edge insertions efficiently, achieving $O(n^2 + m)$ time for the full parameter range $0 \leq \ell < 2k$. The core technical contribution is a detailed implementation for the general range, plus optimized variants for the case $0 \leq \ell \leq k$ and for the unweighted problem, all with careful space-time trade-offs. These results enable faster solutions for rigidity-related problems (e.g., minimum-weight redundantly rigid and globally rigid subgraphs) and have practical impact through an online implementation and applications in framework enumeration and kinematic joint recognition.

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.

Figures (2)

• Figure 1: Impact of inserting edge $uv$ on the $(2, 3)$-components.
• Figure 2: Counterexample to the generalized bounded property.

Theorems & Definitions (33)

• Theorem 1: Lorea, 1979 lorea
• Lemma 1
• proof
• Proposition 1
• proof
• Lemma 2
• proof
• Corollary 1
• proof
• Corollary 2