Thin Trees for Near Minimum Cuts

2026-05-12

AI Summary

This work investigates the existence of spanning trees with $O(1/k)$-sparsity for all near-minimum cuts in $k$-edge-connected graphs. Focusing on the family of cuts whose values do not exceed $(1 + 1/40)k$, the paper establishes, for the first time, that the strong sparse tree conjecture holds over this subset and provides a polynomial-time algorithm to construct such trees. The approach builds upon the polygon representation framework introduced by Benczúr and Goemans, transforming the structure of near-minimum cuts into a layered family problem amenable to efficient resolution. This yields the first deterministic algorithm capable of efficiently constructing $O(1/k)$-sparse spanning trees within this near-minimum cut regime.

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 $η$-near minimum cuts of the graph for $η= 1/40$, 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 Benczúr 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.

Problem

• thin trees
• near minimum cuts
• k-edge-connected graphs
• spanning tree
• strong thin tree conjecture

Innovation

• thin spanning tree
• near minimum cuts
• laminar family
• polygon representation
• graph connectivity