Constant-time algorithms for sparsity matroids

• Authors: Hiro Ito;Shin-Ichi Tanigawa;Yuichi Yoshida
• Venue: ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
• Year: 2012

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. 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-connected-orientable if there exists an orientation $\vec{G}$ of G with a vertex r∈V such that $\vec{G}$ 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 any k≥2 and (proper) ℓ≥0, every one-sided error tester for (k,ℓ)-fullness and (k,ℓ)-edge-connected-orientability requires Ω(n) queries.