Approximate minimum cuts and their enumeration

TL;DR

The paper proves that approximate minimum cuts in a connected graph are uniquely characterized by certain terminal cuts, enabling efficient enumeration of all such cuts for fixed approximation factors.

Contribution

It establishes a novel characterization of approximate minimum cuts as unique terminal cuts, providing a new proof for their bounded number and polynomial-time enumeration.

Findings

Number of α-approximate minimum cuts is n^{O(α)}.

All approximate minimum cuts can be enumerated in polynomial time for fixed α.

Provides an alternative proof for the enumeration bound.

Abstract

We show that every $\alpha$-approximate minimum cut in a connected graph is the unique minimum $(S,T)$-terminal cut for some subsets $S$ and $T$ of vertices each of size at most $\lfloor2\alpha\rfloor+1$. This leads to an alternative proof that the number of $\alpha$-approximate minimum cuts in a $n$-vertex connected graph is $n^{O(\alpha)}$ and they can all be enumerated in deterministic polynomial time for constant $\alpha$.