Asymptotically Faster Algorithms for Recognizing $(k,\ell)$-Sparse Graphs
For fixed integers $k,\ell$ with $0\leq\ell<3k$, we study the problem of deciding whether a graph is $(k,\ell)$-sparse, a notion arising in combinatorial optimization and rigidity theory. Standard pebble-game algorithms take $O(n^2)$ time for $0\leq\ell<2k$ and $O(n^3)$ time for $2k\leq\ell<3k$, although faster algorithms were known for some parameter choices. We develop efficient recognition algorithms using bounded-indegree orientations, reductions to rooted arc-connectivity, augmenting paths, and centroid decomposition. For $0\leq\ell<2k$, they run in $O(n^{1+o(1)})$ time, yielding the first subquadratic bound valid throughout the entire range. Purely combinatorial implementations run in $O(n\sqrt n)$ time throughout this range. For graphs with $O(n)$ edges, we also improve the purely combinatorial running time for decomposing the edge set into a fixed number of forests, or certifying that no such decomposition exists, from $O(n\sqrt{n\log n})$ to $O(n\sqrt n)$. For simple graphs with $2k\leq\ell<3k$, the general recognition algorithm runs in $O(n^2)$ time when $\ell\leq2k+2$ and in $O(n^2\log n)$ time otherwise. When $\ell=2k$ and $k\leq3$, a specialized algorithm improves this to $O(n^{1+o(1)})$ and admits a purely combinatorial $O(n\sqrt n)$ implementation. Whenever the input graph is not $(k,\ell)$-sparse, our recognition algorithms return an explicit violating vertex set.