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

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.