Graph Sparsification by Edge-Connectivity and Random Spanning Trees

We present new approaches to constructing graph sparsifiers — weighted subgraphs for which every cut has the same value as the original graph, up to a factor of (1 ± ǫ). Our first approach independently samples each edge uv with probability inversely proportional to the edge-connectivity between u and v. The fact that this approach produces a sparsifier resolves a question posed by Benczúr and Karger (2002). Concurrent work of Hariharan and Panigrahi also resolves this question. Our second approach constructs a sparsifier by forming the union of several uniformly random spanning trees. Both of our approaches produce sparsifiers with O(n log(n)/ǫ) edges. Our proofs are based on extensions of Karger’s contraction algorithm, which may be of independent interest.