[Paper Review] Analogs of the van der Waerden and Tverberg conjectures for haffnians

Shmuel Friedland|arXiv (Cornell University)|Feb 12, 2011

Markov Chains and Monte Carlo Methods24 references3 citations

TL;DR

This paper introduces analogs of the van der Waerden and Tverberg conjectures for haffnians, focusing on symmetric doubly stochastic matrices with zero diagonal (denoted Ψ₂ₙ). It establishes that the uniform matrix 1/(2n−1)A(K₂ₙ) is a local minimizer of the k-haffnian functional on Ψ₂ₙ for k=2, proving the conjecture holds in this case, though it fails for k=n when n is large. The key contribution is a lower bound for hafₖ(B) over Ψ₂ₙ using hyperbolicity and mixed derivative inequalities.

ABSTRACT

We discuss here analogs of van der Waerden and Tverberg permanent conjectures for haffnians on the convex set of matrices whose extreme points are symmetric permutation matrices with zero diagonal.

Motivation & Objective

• To extend the van der Waerden and Tverberg conjectures—originally for permanents of doubly stochastic matrices—to haffnians of symmetric matrices with zero diagonal.
• To define and characterize the convex set Ψ₂ₙ of symmetric doubly stochastic matrices with zero diagonal, whose extreme points are symmetric permutation matrices with zero diagonal.
• To establish lower bounds for the k-haffnian functional hafₖ(B) over Ψ₂ₙ, particularly investigating whether the uniform matrix 1/(2n−1)A(K₂ₙ) minimizes hafₖ(B).
• To analyze the critical point structure of hafₖ(B) on Ψ₂ₙ using differential geometry and hyperbolic polynomial theory.

Proposed method

• Define the convex set Ψ₂ₙ as the set of symmetric, doubly stochastic matrices with zero diagonal, characterized by edge-odd-set inequalities (1.2).
• Use the haffnian hafₖ(B) as the sum of weighted k-matchings in a complete graph K₂ₙ with edge weights given by B.
• Apply the theory of hyperbolic polynomials, particularly the mixed derivative inequality from Gurvits (2009), to derive lower bounds for hafₖ(B).
• Use the identity hafₖ(F) = (1/k)∑_{i<j} f_{ij} haf_{k−1}(F[[2n]\{i,j}]) to expand hafₖ(B+Y) in a Taylor series around the uniform matrix C=1/(2n−1)A(K₂ₙ).
• Show that the linear term in the expansion vanishes due to the constraint Y1=0, implying C is a critical point.
• Prove the Hessian of hafₖ at C is positive definite on the tangent space Φ₂ₙ, confirming C is a local minimizer for k=2.

Experimental results

Research questions

• RQ1 Is the uniform matrix 1/(2n−1)A(K₂ₙ) the minimizer of the k-haffnian functional hafₖ(B) over the convex set Ψ₂ₙ for all k=2,…,n?
• RQ2 Does the conjecture hafₖ(B) ≥ hafₖ(1/(2n−1)A(K₂ₙ)) hold for all B∈Ψ₂ₙ, with equality only at the uniform matrix?
• RQ3 For which values of k and n does the conjecture fail, and what are the structural reasons for its failure?
• RQ4 Can hyperbolic polynomial theory be used to derive nontrivial lower bounds for hafₖ(B) over Ψ₂ₙ?
• RQ5 What is the asymptotic behavior of the minimal k-haffnian value μₖ,ₙ as n→∞?

Key findings

• The conjecture hafₖ(B) ≥ hafₖ(1/(2n−1)A(K₂ₙ)) holds for k=2, as the uniform matrix is a strict local minimizer of haf₂(B) on Ψ₂ₙ.
• For B∈Ψ₂ₙ with exactly one positive eigenvalue, the bound hafₙ(B) ≥ (n−1)ⁿ⁽ⁿ⁻¹⁾ / nⁿ⁽ⁿ⁻¹⁾ ≈ e⁻ⁿ√e holds, with asymptotic approximation (2.6) for large n.
• For k=2,…,n−1, the lower bound (2.7) is derived using Theorem 3.1 from Gurvits (2009) on mixed derivatives of hyperbolic polynomials.
• The Hessian of hafₖ at the uniform matrix C is positive definite on the tangent space Φ₂ₙ, confirming C is a strict local minimizer for all k=2,…,n.
• The conjecture fails for k=n and large n, as shown by structural counterexamples in the paper’s analysis.
• The value hafₖ(1/(2n−1)A(K₂ₙ)) is explicitly computed as (1/(2n−1)ᵏ) times the number of k-matchings in K₂ₙ, which is a known combinatorial quantity.