Sharp Bounds on the Number of Small Cuts

2026-09-09

AI Summary

本文通过结合分裂、采样和嵌套顶点集大小的界限方法，解决了在n-顶点无向多重图中估计小于特定倍数最小割值的切割数量问题。

Abstract

Let $λ$ be the minimum cut value of an $n$-vertex undirected multigraph. For every fixed $α>1$, we prove that there are $O(n^{\lceil2α\rceil-1})$ cuts of size strictly below $αλ$. The exponent is sharp. The proof combines splitting off and sampling with a bound on the size of nested families of vertex sets.

Problem

• minimum cut value
• undirected multigraph
• sharp bounds

Innovation

• sharp bounds
• minimum cut value
• splitting off
• sampling
• nested families