[Paper Review] Sums of permanental minors using Grassmann algebra

P. Butera, M. Pernici|arXiv (Cornell University)|Jun 20, 2014

Advanced Topics in Algebra3 citations

TL;DR

This paper presents a Grassmann algebra-based algorithm that computes the generating function for sums of permanental minors of an n×n matrix in O(2ⁿn³) time, significantly improving upon the Brualdi-Ryser formula’s O(2^{5n/2}) complexity. The method leverages Grassmann integration and anticommuting variables to efficiently compute matching and independence polynomials for graphs, with optimized performance for banded matrices and verified results on lattice systems including dimer models and fullerenes.

ABSTRACT

We show that a formalism proposed by Creutz to evaluate Grassmann integrals provides an algorithm of complexity $O(2^n n^3)$ to compute the generating function for the sum of the permanental minors of a matrix of order $n$. This algorithm improves over the Brualdi-Ryser formula, whose complexity is at least $O(2^{\frac{5n}{2}})$. In the case of a banded matrix with band width $w$ and rank $n$ the complexity is $O(2^{min(2w, n)} (w + 1) n^2)$. Related algorithms for the matching and independence polynomials of graphs are presented.

Motivation & Objective

• To develop a more efficient algorithm for computing the sum of permanental minors of a matrix, which is central to calculating matching and independence polynomials in statistical physics and combinatorics.
• To overcome the exponential complexity of existing methods like the Brualdi-Ryser formula, especially for large or structured matrices.
• To apply Grassmann algebra formalism to graph-theoretic problems such as dimer coverings and independent sets, enabling efficient computation via Berezin integration.
• To provide a practical, open-source Python implementation for computing matching and independence polynomials, with performance benchmarks on lattices and fullerenes.
• To resolve discrepancies in prior Witten index calculations for supersymmetric dimer models on square and hexagonal lattices by providing corrected values.

Proposed method

• The paper uses a Grassmann algebra formalism where each hard object (e.g., dimer or independent vertex set) is represented by a product of nilpotent, commuting elements ηᵢ = θᵢθ̄ᵢ derived from anticommuting variables.
• The generating function for hard object configurations is expressed as a Grassmann integral ⟨∏ₐ(1 + wₐ∏ᵢ∈Vₐηᵢ)⟩, with the Berezin integral extracting the sum of monomials corresponding to non-overlapping configurations.
• The algorithm simplifies Creutz’s Grassmann integration method into a sequence of polynomial manipulations, reducing computational complexity by optimizing variable ordering and monomial reduction.
• For banded matrices with bandwidth w, the complexity reduces to O(2^{min(2w,n)}(w+1)n²), achieving polynomial scaling when w is fixed.
• The method is generalized to compute both matching polynomials (via permanental minors of reduced adjacency matrices) and independence polynomials (via vertex-based hard objects).
• An efficient implementation in Python, named 'hobj', is provided, using optimized polynomial arithmetic and avoiding full polynomial construction when only the sum of coefficients is needed.

Experimental results

Research questions

• RQ1 Can Grassmann integration be systematically applied to compute the sum of permanental minors of a matrix more efficiently than existing combinatorial formulas?
• RQ2 What is the computational complexity of computing the matching generating polynomial for general and banded matrices using Grassmann algebra?
• RQ3 How does the proposed algorithm compare with prior results for the Witten index in supersymmetric dimer models on square and hexagonal lattices?
• RQ4 Can the Grassmann formalism be extended to compute the independence polynomial of a graph, and how does its performance compare to matching polynomial computation?
• RQ5 To what extent can variable ordering and monomial reduction strategies improve the practical performance of Grassmann-based enumeration algorithms?

Key findings

• The proposed algorithm computes the sum of permanental minors in O(2ⁿn³) time, a significant improvement over the Brualdi-Ryser formula’s O(2^{5n/2}) complexity.
• For banded matrices with bandwidth w, the complexity reduces to O(2^{min(2w,n)}(w+1)n²), achieving polynomial scaling when w is fixed and n grows.
• The algorithm correctly computes the Witten index W for square and hexagonal lattices, resolving discrepancies with prior results that likely arose from inadequate large-integer arithmetic.
• For the 10×10 square lattice, the Witten index is computed as 323,585,192,7936, which differs from the value reported in Ref.[Eerten] by a factor of ~1000, with the correct value lying in the 1.333 range for |W|^{1/mn}.
• The independence polynomial of the C₆₀ fullerene (Buckminsterfullerene) is computed as having a sum of coefficients equal to 217,727,997,152, matching known results.
• The Python module 'hobj' enables fast computation of matching and independence polynomials, with the sum of coefficients for the 20×20 banded matrix computed as 11,936,810,897,247,956,264,161,397,956,481,650,508,142,206,788 in under a second on a standard PC.