Constant-Time Algorithms for Sparsity Matroids

A graph G = (V,E) is called (k, l)-full if G contains a subgraph H = (V, F ) of k|V |− l edges such that, for any non-empty F ′ ⊆ F , |F ′| ≤ k|V (F ′)| − l holds. Here, V (F ) denotes the set of vertices incident to F . It is known that the family of edge sets of (k, l)-full graphs forms a family of matroid, known as the sparsity matroid of G. In this paper, we give a constant-time approximation algorithm for the rank of the sparsity matroid of a degree-bounded undirected graph. This leads to a constant-time tester for (k, l)-fullness in the bounded-degree model, (i.e., we can decide with high probability whether an input graph satisfies a property P or far from P ). Depending on the values of k and l, it can test various properties of a graph such as connectivity, rigidity, and how many spanning trees can be packed. Based on this result, we also propose a constant-time tester for (k, l)-edge-connected-orientability in the bounded-degree model, where an undirected graph G is called (k, l)-edge-connectedorientable if there exists an orientation ~ G of G with a vertex r ∈ V such that ~ G contains k arc-disjoint dipaths from r to each vertex v ∈ V and l arc-disjoint dipaths from each vertex v ∈ V to r. A tester is called a one-sided error tester for P if it always accepts a graph satisfying P . We show, for k ≥ 2 and (proper) l ≥ 0, any one-sided error tester for (k, l)-fullness and (k, l)-edgeconnected-orientability requires Ω(n) queries.