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From Trees to Polynomials and Back Again: New Capacity Bounds with Applications to TSP

Leonid Gurvits, Nathan Klein, Jonathan Leake

TL;DR

The paper develops robust, degree-free bounds for real stable polynomials and strongly Rayleigh distributions by reducing worst-case inputs to sparse, forest-structured matrices via a generalized productization and extreme-point analysis. A key insight is that extreme matrices correspond to bipartite forests, enabling an induction on leaves that yields strong capacity and coefficient bounds independent of total degree. These results feed into two main applications: a tight, capacity-based understanding of the permanent under constrained row/column sums, and a significant refinement of probabilistic bounds used in a metric TSP approximation algorithm, achieving a near-3/2 approximation with both randomized and deterministic incarnations. Overall, the work unifies polynomial capacity, combinatorial structure, and probabilistic analysis to improve bounds and streamline analyses across combinatorics and TCS.

Abstract

We give simply exponential lower bounds on the probabilities of a given strongly Rayleigh distribution, depending only on its expectation. This resolves a weak version of a problem left open by Karlin-Klein-Oveis Gharan in their recent breakthrough work on metric TSP, and this resolution leads to a minor improvement of their approximation factor for metric TSP. Our results also allow for a more streamlined analysis of the algorithm. To achieve these new bounds, we build upon the work of Gurvits-Leake on the use of the productization technique for bounding the capacity of a real stable polynomial. This technique allows one to reduce certain inequalities for real stable polynomials to products of affine linear forms, which have an underlying matrix structure. In this paper, we push this technique further by characterizing the worst-case polynomials via bipartitioned forests. This rigid combinatorial structure yields a clean induction argument, which implies our stronger bounds. In general, we believe the results of this paper will lead to further improvement and simplification of the analysis of various combinatorial and probabilistic bounds and algorithms.

From Trees to Polynomials and Back Again: New Capacity Bounds with Applications to TSP

TL;DR

The paper develops robust, degree-free bounds for real stable polynomials and strongly Rayleigh distributions by reducing worst-case inputs to sparse, forest-structured matrices via a generalized productization and extreme-point analysis. A key insight is that extreme matrices correspond to bipartite forests, enabling an induction on leaves that yields strong capacity and coefficient bounds independent of total degree. These results feed into two main applications: a tight, capacity-based understanding of the permanent under constrained row/column sums, and a significant refinement of probabilistic bounds used in a metric TSP approximation algorithm, achieving a near-3/2 approximation with both randomized and deterministic incarnations. Overall, the work unifies polynomial capacity, combinatorial structure, and probabilistic analysis to improve bounds and streamline analyses across combinatorics and TCS.

Abstract

We give simply exponential lower bounds on the probabilities of a given strongly Rayleigh distribution, depending only on its expectation. This resolves a weak version of a problem left open by Karlin-Klein-Oveis Gharan in their recent breakthrough work on metric TSP, and this resolution leads to a minor improvement of their approximation factor for metric TSP. Our results also allow for a more streamlined analysis of the algorithm. To achieve these new bounds, we build upon the work of Gurvits-Leake on the use of the productization technique for bounding the capacity of a real stable polynomial. This technique allows one to reduce certain inequalities for real stable polynomials to products of affine linear forms, which have an underlying matrix structure. In this paper, we push this technique further by characterizing the worst-case polynomials via bipartitioned forests. This rigid combinatorial structure yields a clean induction argument, which implies our stronger bounds. In general, we believe the results of this paper will lead to further improvement and simplification of the analysis of various combinatorial and probabilistic bounds and algorithms.
Paper Structure (42 sections, 43 theorems, 124 equations)

This paper contains 42 sections, 43 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Application: Minimum Permanent
  4. Application: Metric TSP
  5. Technical Overview
  6. Notation.
  7. Conceptual strategy.
  8. Conceptual Strategy, in More Detail
  9. Step 1: Reduce to products of affine linear forms via productization.
  10. Step 2: Reduce to extreme points.
  11. Step 3: Extreme points correspond to bipartitioned forests.
  12. Step 4: Induction on leaf vertices of the bipartitioned forests.
  13. Some comments on tightness of the bounds.
  14. Example: The Univariate Case
  15. Proof of Application: Metric TSP
  16. ...and 27 more sections

Key Result

Theorem 2.1

Let $p \in \mathbb{R}_{\geq 0}[x_1,\ldots,x_n]$ be a real stable polynomial in $n$ variables, and fix any $\bm\kappa \in \mathbb{Z}^n$ with non-negative entries. If $p(\bm{1}) = 1$ and $\|\bm\kappa - \nabla p(\bm{1})\|_1 < 1$, then This bound is tight for any fixed $\bm\kappa$ with strictly positive entries.

Theorems & Definitions (71)

  • Theorem 2.1: Main non-homogeneous capacity bound
  • Corollary 2.2: Main non-homogeneous coefficient bound
  • Theorem 2.3
  • Proposition 2.4
  • Theorem 2.5: Prop. 5.1 of KKO21
  • Theorem 2.6: Improved probability lower bound
  • Theorem 2.7
  • Theorem 3.1: = \ref{['thm:main_nh']} and \ref{['cor:main_nh_coeff']}
  • Theorem 4.1: Strongest form of the probability bound
  • proof
  • ...and 61 more