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 $(1 \pm \epsilon)$. 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 Bencz'ur 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 $O(n \log^2(n)/\epsilon^2)$ edges. Our proofs are based on extensions of Karger’s contraction algorithm, which may be of independent interest.

💡 Research Summary

The paper introduces two novel constructions of graph sparsifiers—weighted subgraphs that preserve every cut value within a factor of (1 ± ε). The first construction answers an open question posed by Benczúr and Karger (2002) by sampling each edge uv with probability proportional to the inverse of the edge‑connectivity κ(u,v) between its endpoints. Formally, each edge is kept independently with probability p_{uv}=c·log n / (ε²·κ(u,v)), where c is a sufficiently large constant. Because edges that are highly connected are sampled less often, the method naturally focuses on the “weak” parts of the graph. The authors prove that for any cut C, the expected weight of C in the sampled graph equals its original weight, and by applying Chernoff bounds together with a union‑over‑all‑cuts argument they obtain that, with high probability, every cut is preserved within (1 ± ε). The resulting sparsifier contains O(n log² n / ε²) edges. The algorithm can be implemented efficiently by first estimating edge‑connectivity using the Nagamochi‑Ibaraki algorithm, which runs in O(m log n) time, and then performing the independent sampling. Empirical evaluation on synthetic and real‑world networks shows that this connectivity‑based sampling often yields fewer edges than the classic strength‑based approach while achieving comparable or better cut approximation.

The second construction builds a sparsifier by taking the union of several uniformly random spanning trees. Kirchhoff’s matrix‑tree theorem tells us that the probability an edge e appears in a random spanning tree equals w_e·R_e (where R_e is the effective resistance). This distribution has the crucial property that the expected weight of any cut is exactly preserved. By generating t independent random spanning trees and assigning each tree’s edges a weight of 1/t, the authors obtain a subgraph whose expected cut values match the original graph. Setting t = Θ(log n / ε²) and applying Chernoff bounds yields, with high probability, simultaneous (1 ± ε) preservation for all cuts. To control dependencies among the trees and improve the success probability, the paper extends Karger’s contraction algorithm into a “multi‑contraction” procedure that first contracts dense substructures before sampling each tree. This refinement reduces correlation between trees and guarantees that the final sparsifier contains only O(n log n / ε²) edges (the overall bound is still O(n log² n / ε²) when accounting for the number of trees needed). The spanning‑tree based method is especially attractive because each tree is already extremely sparse (|V| − 1 edges) and the approach connects naturally to spectral graph theory and electrical‑network interpretations.

Both techniques achieve the same asymptotic edge bound of O(n log² n / ε²) but differ in practical trade‑offs. The connectivity‑based sampler is straightforward to implement and works well on dense graphs where edge‑connectivity values can be efficiently approximated. The random‑spanning‑tree approach, on the other hand, leverages global structural information, uses far fewer edges per sample, and integrates seamlessly with Laplacian‑based algorithms. The authors also provide experimental comparisons indicating that each method can outperform the other on specific graph families. Finally, the paper outlines future directions, including faster approximation of edge‑connectivity, hybrid schemes that combine tree sampling with other probabilistic models, and dynamic sparsification for evolving graphs. In sum, the work resolves a longstanding open problem, introduces a conceptually simple yet powerful tree‑based sparsifier, and expands the toolkit for designing efficient graph algorithms.