Constant-Time Algorithms for Sparsity Matroids

Hiro Ito, Shin-ichi Tanigawa, Yuichi Yoshida

Abstract: A graph G = (V, E) is called (k, ℓ)-sparse if |F| ≤ k|V(F)| − ℓ for any F ⊆ E with F ≠ ∅. Here, V(F) denotes the set of vertices incident to F. A graph G = (V,E) is called (k,ℓ)-full if G contains a (k,ℓ)-sparse subgraph with |V| vertices and k|V| − ℓ edges. The family of edge sets of (k,ℓ)-sparse subgraphs forms a family of independent sets of a matroid on E, known as the sparsity matroid of G. In this paper, we give a constant-time algorithm that approximates the rank of the sparsity matroid associated with a degree-bounded undirected graph. This algorithm leads to a constant-time tester for (k,ℓ)-fullness in the bounded-degree model, (i.e., we can decide with high probability whether the input graph satisfies a property or far from it). Depending on the values of k and ℓ, our algorithm can test various properties of graphs such as connectivity, rigidity, and how many spanning trees can be packed in a unified manner.