[Paper Review] Approximating the Permanent via Nonabelian Determinants

Cristopher Moore, Alexander Russell|ArXiv.org|Jun 9, 2009

Markov Chains and Monte Carlo Methods7 references22 citations

TL;DR

This paper investigates algebraic approaches to approximating the permanent of a {0,1} matrix using determinants over noncommutative algebras, particularly d×d matrix algebras and group algebras. It shows that while unsymmetrized determinants yield a critical ratio of (1+O(1/d))^n, the symmetrized determinant estimator has a critical ratio of Ω(2^n / n^O(d)), implying significant obstacles for polynomial-time approximation schemes using this method.

ABSTRACT

Celebrated work of Jerrum, Sinclair, and Vigoda has established that the permanent of a {0,1} matrix can be approximated in randomized polynomial time by using a rapidly mixing Markov chain. A separate strand of the literature has pursued the possibility of an alternate, purely algebraic, polynomial-time approximation scheme. These schemes work by replacing each 1 with a random element of an algebra A, and considering the determinant of the resulting matrix. When A is noncommutative, this determinant can be defined in several ways. We show that for estimators based on the conventional determinant, the critical ratio of the second moment to the square of the first--and therefore the number of trials we need to obtain a good estimate of the permanent--is (1 + O(1/d))^n when A is the algebra of d by d matrices. These results can be extended to group algebras, and semi-simple algebras in general. We also study the symmetrized determinant of Barvinok, showing that the resulting estimator has small variance when d is large enough. However, for constant d--the only case in which an efficient algorithm is known--we show that the critical ratio exceeds 2^{n} / n^{O(d)}. Thus our results do not provide a new polynomial-time approximation scheme for the permanent. Indeed, they suggest that the algebraic approach to approximating the permanent faces significant obstacles. We obtain these results using diagrammatic techniques in which we express matrix products as contractions of tensor products. When these matrices are random, in either the Haar measure or the Gaussian measure, we can evaluate the trace of these products in terms of the cycle structure of a suitably random permutation. In the symmetrized case, our estimates are then derived by a connection with the character theory of the symmetric group.

Motivation & Objective

• To evaluate the feasibility of polynomial-time approximation schemes for the permanent using algebraic estimators based on nonabelian determinants.
• To analyze the variance of determinant-based estimators when matrix entries are replaced with random elements from noncommutative algebras.
• To compare the performance of unsymmetrized versus symmetrized determinants in terms of the critical ratio (second moment over square of first moment).
• To determine whether algebraic methods can achieve subexponential or polynomial-time approximation of the permanent, especially when the algebra dimension d is small.
• To investigate the role of group algebras and semi-simple algebras in constructing low-variance estimators for the permanent.

Proposed method

• Uses diagrammatic techniques to express matrix products as contractions of tensor products, enabling trace evaluation via cycle structures of random permutations.
• Applies Haar and Gaussian measures to random matrices in noncommutative algebras, computing expectations of products using combinatorial properties of permutations.
• Relies on character theory of the symmetric group to analyze the symmetrized determinant estimator, particularly in the case of large d.
• Defines two types of estimators: the conventional determinant (non-symmetrized) and Barvinok’s symmetrized determinant, both used to estimate the permanent via random matrix substitution.
• Computes the critical ratio E[X²]/E[X]² for both estimators, using trace and norm-based moment analysis, and relates Frobenius norm estimators to trace-squared estimators via diagram rewiring.
• Derives bounds on the critical ratio by analyzing loop structures in tensor contraction diagrams, showing that rewiring affects loop count by at most one per move.

Experimental results

Research questions

• RQ1 Can the permanent of a {0,1} matrix be approximated in polynomial time using determinants over noncommutative algebras?
• RQ2 How does the critical ratio of determinant-based estimators scale with the dimension d of the algebra, particularly for d=O(1) versus growing d?
• RQ3 Does the symmetrized determinant estimator proposed by Barvinok yield a polynomial-time approximation scheme for the permanent?
• RQ4 What is the variance behavior of algebraic estimators when matrix entries are drawn from d×d matrix algebras or group algebras?
• RQ5 Can diagrammatic and group-theoretic methods be used to bound higher moments and establish concentration of measure for determinant-based estimators?

Key findings

• For the unsymmetrized determinant estimator over d×d matrix algebras, the critical ratio is (1+O(1/d))^n, which becomes subexponential as d increases.
• For the symmetrized determinant estimator with d=O(1), the critical ratio is Ω(2^n / n^O(d)), indicating exponential variance even when the estimator can be computed efficiently.
• The critical ratio for the symmetrized estimator exceeds 2^n / n^O(d) when d is constant, suggesting that such estimators are not suitable for polynomial-time approximation schemes.
• The Frobenius norm estimator X_Frob has moments bounded within a factor of d^2 of the trace-squared estimator, implying similar variance scaling under worst-case bounds.
• Diagrammatic rewiring techniques show that changing from trace-squared to Frobenius norm estimators alters loop counts by at most one per move, justifying the d^2 factor in moment bounds.
• The results suggest that algebraic approaches to approximating the permanent face fundamental obstacles, especially when d is constant, and that higher-dimensional algebras are necessary for improved concentration.