Thin Trees for Near Minimum Cuts

TL;DR

This paper proves that every sufficiently connected graph has a spanning tree thin on near minimum cuts, advancing the strong thin tree conjecture under specific conditions, and provides a polynomial-time construction method.

Contribution

It demonstrates the conjecture holds for near minimum cuts with a polynomial-time construction, using the polygon representation and laminar family techniques.

Findings

Proves the strong thin tree conjecture for near minimum cuts with η=1/40.

Constructs such a thin spanning tree in polynomial time.

Reduces the problem to finding a tree thin for laminar families.

Abstract

The strong thin tree conjecture states that every $k$-edge-connected graph $G$ contains an $O(1/k)$-thin spanning tree, meaning a spanning tree which contains at most an $O(1/k)$ fraction of the edges across each cut in $G$. This conjecture is still open despite significant effort; the best current result by Anari and Oveis Gharan shows the existence of an $O(\text{polyloglog}(n)/k)$-thin tree. In this work, we demonstrate that the conjecture is true if one only requires thinness for the set of $\eta$-near minimum cuts of the graph for $\eta = 1/40$, in other words, for the set of cuts with fewer than $(1+1/40)k$ edges. Our approach constructs such a tree in polynomial time. To show this, we utilize the structure of near minimum cuts, and in particular the polygon representation of Bencz\'ur and Goemans, to reduce to the previously solved problem of finding a spanning tree that is $O(1/k)$-thin for all sets in a laminar family.