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Thin Trees via $k$-Respecting Cut Identities

Mohit Daga

TL;DR

This work tackles certifying the thinness of spanning trees by introducing k-respecting cut identities, which express cuts crossing a tree in at most $k$ edges in terms of pairwise statistics. It provides a polynomial-time procedure to compute an exact k-thinness certificate Θ_k(T) for fixed $k$, along with a witnessing cut, after $O(n^2)$ preprocessing, and an end-to-end ensemble certification via sampling from a fractional tree packing. The framework enables a tree-local oracle that evaluates all $k$-respecting cuts, supports exact and approximate evaluations, and yields per-cut certificates for near-minimum cuts when combined with sampling and counting results. In planar and bounded-genus graphs, duality and dual girth give sharper certified bounds, showing the method’s compatibility with structural graph classes. Overall, the paper advances the constructive thin-tree program by turning thinness into verifiable certificates and suggesting practical pathways to stitching ensembles into a single certifiable tree.

Abstract

Thin spanning trees lie at the intersection of graph theory, approximation algorithms, and combinatorial optimization. They are central to the long-standing \emph{thin tree conjecture}, which asks whether every $k$-edge-connected graph contains an $O(1/k)$-thin tree, and they underpin algorithmic breakthroughs such as the $O(\log n/\log\log n)$-approximation for ATSP. Yet even the basic algorithmic task of \emph{verifying} that a given tree is thin has remained elusive: checking thinness requires reasoning about exponentially many cuts, and no efficient certificates have been known. We introduce a new machinery of \emph{$k$-respecting cut identities}, which express the weight of every cut that crosses a spanning tree in at most $k$ edges as a simple function of pairwise ($2$-respecting) cuts. This yields a tree-local oracle that, after $O(n^2)$ preprocessing, evaluates such cuts in $O_k(1)$ time. Building on this oracle, we give the first procedure to compute the exact $k$-thinness certificate $Θ_k(T)$ of any spanning tree for fixed $k$ in time $\tilde O(n^2+n^k)$, outputting both the certificate value and a witnessing cut. Beyond general graphs, our framework yields sharper guarantees in structured settings. In planar graphs, duality with cycles and dual girth imply that every spanning tree admits a verifiable certificate $Θ_k(T)\le k/λ$ (hence $O(1/λ)$ for constant $k$). In graphs embedded on a surface of genus $γ$, refined counting gives certified (per-cut) bounds $O((\log n+γ)/λ)$ via the same ensemble coverage.

Thin Trees via $k$-Respecting Cut Identities

TL;DR

This work tackles certifying the thinness of spanning trees by introducing k-respecting cut identities, which express cuts crossing a tree in at most edges in terms of pairwise statistics. It provides a polynomial-time procedure to compute an exact k-thinness certificate Θ_k(T) for fixed , along with a witnessing cut, after preprocessing, and an end-to-end ensemble certification via sampling from a fractional tree packing. The framework enables a tree-local oracle that evaluates all -respecting cuts, supports exact and approximate evaluations, and yields per-cut certificates for near-minimum cuts when combined with sampling and counting results. In planar and bounded-genus graphs, duality and dual girth give sharper certified bounds, showing the method’s compatibility with structural graph classes. Overall, the paper advances the constructive thin-tree program by turning thinness into verifiable certificates and suggesting practical pathways to stitching ensembles into a single certifiable tree.

Abstract

Thin spanning trees lie at the intersection of graph theory, approximation algorithms, and combinatorial optimization. They are central to the long-standing \emph{thin tree conjecture}, which asks whether every -edge-connected graph contains an -thin tree, and they underpin algorithmic breakthroughs such as the -approximation for ATSP. Yet even the basic algorithmic task of \emph{verifying} that a given tree is thin has remained elusive: checking thinness requires reasoning about exponentially many cuts, and no efficient certificates have been known. We introduce a new machinery of \emph{-respecting cut identities}, which express the weight of every cut that crosses a spanning tree in at most edges as a simple function of pairwise (-respecting) cuts. This yields a tree-local oracle that, after preprocessing, evaluates such cuts in time. Building on this oracle, we give the first procedure to compute the exact -thinness certificate of any spanning tree for fixed in time , outputting both the certificate value and a witnessing cut. Beyond general graphs, our framework yields sharper guarantees in structured settings. In planar graphs, duality with cycles and dual girth imply that every spanning tree admits a verifiable certificate (hence for constant ). In graphs embedded on a surface of genus , refined counting gives certified (per-cut) bounds via the same ensemble coverage.
Paper Structure (32 sections, 22 theorems, 61 equations, 1 figure, 1 algorithm)

This paper contains 32 sections, 22 theorems, 61 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Thin trees and their conjectured power
  3. Packing and cut structure
  4. Our contribution and conceptual message
  5. Preliminaries and Results
  6. Thinness and its algorithmic role
  7. Our angle.
  8. The cut space and symmetric difference
  9. Planar and bounded-genus graphs
  10. Main theorems
  11. Organization of the paper
  12. Evaluation of $k$-Respecting Cuts
  13. $k$-Thinness: Certificates, Coverage, and Global Guarantee
  14. The $k$-thinness certificate and exact evaluation
  15. Coverage of near-minimum cuts by sampled trees
  16. ...and 17 more sections

Key Result

theorem 2.1

Let $G=(V,E, w)$ be a weighted graph and let $T$ be a rooted spanning tree. For any cut $A \subseteq V$ with $|E(T)\cap \delta(A)| \le k$, the cut value $w(\delta(A))$ can be expressed in closed form using only pairwise quantities of the form $w(\delta(D_T(u)\oplus D_T(v)))$. In particular, after $O

Figures (1)

  • Figure 1: Ancestor–descendant configurations for the four cases used in the proof of Lemma \ref{['lem:pairwise-sufficiency']}.

Theorems & Definitions (37)

  • theorem 2.1: Exact evaluation of $k$-respecting cuts
  • theorem 2.2: Exact evaluation of $k$-thinness
  • theorem 2.3: Near-min cuts become $k$-respecting
  • theorem 2.4: Ensemble coverage $\Rightarrow$ per-cut certificates
  • theorem 2.5: End-to-end ensemble certification
  • lemma 3.1: Exact $k$--respecting cut evaluation from pairwise data
  • proof
  • lemma 3.2: Pairwise sufficiency
  • remark 3.3: Per-cut evaluation time
  • lemma 3.4: Weighted/capacitated extension
  • ...and 27 more