[Paper Review] A Deterministic Approximation Algorithm for Computing a Permanent of a 0,1 matrix

David Gamarnik, Dmitriy Katz|ArXiv.org|Feb 2, 2007

Markov Chains and Monte Carlo Methods8 references4 citations

TL;DR

This paper presents a deterministic approximation algorithm for computing the permanent of a 0-1 matrix within a multiplicative factor of $(1+\epsilon)^n$. It leverages a deterministic algorithm for counting partial matchings and Jerrum-Vazirani expander decomposition, achieving polynomial time for constant-degree expanders and $\exp(O(n^{2/3}\log^3 n))$ time for general matrices.

ABSTRACT

We construct a deterministic approximation algorithm for computing a permanent of a $0,1$ $n$ by $n$ matrix to within a multiplicative factor $(1+ε)^n$, for arbitrary $ε>0$. When the graph underlying the matrix is a constant degree expander our algorithm runs in polynomial time (PTAS). In the general case the running time of the algorithm is $\exp(O(n^{2\over 3}\log^3n))$. For the class of graphs which are constant degree expanders the first result is an improvement over the best known approximation factor $e^n$ obtained in \cite{LinialSamorodnitskyWigderson}. Our results use a recently developed deterministic approximation algorithm for counting partial matchings of a graph Bayati et al., and Jerrum-Vazirani decomposition method.

Motivation & Objective

• To close the gap between randomized and deterministic algorithms for permanent computation in the $\#P$-complete setting.
• To provide a deterministic approximation algorithm with a better multiplicative factor than the prior $e^n$ bound from [LSW00].
• To extend the applicability of deterministic counting techniques to permanent computation via partial matchings and expander decomposition.
• To achieve a polynomial-time approximation for matrices whose underlying graphs are constant-degree expanders.

Proposed method

• The algorithm reduces the permanent computation to estimating the partition function over all partial and full matchings in the graph.
• It uses a deterministic approximation algorithm for counting partial matchings from [BGK+], which runs in polynomial time for constant-degree graphs.
• For general graphs, it applies Jerrum-Vazirani expander decomposition to recursively partition the graph into expanders and non-expanders.
• The permanent is estimated recursively by combining approximations of subgraphs, using the multiplicative factor $(1+\epsilon)^n$ across recursive calls.
• The running time is bounded via recursive analysis, showing $g(n) \leq \exp(O(n^{2/3}\log^3 n))$ using bounds on binomial coefficients and recursive subproblem sizes.
• The algorithm ensures approximation factor $(1+\epsilon)^n$ by maintaining the bound across all recursive subproblems and combining results multiplicatively.

Experimental results

Research questions

• RQ1 Can a deterministic algorithm achieve a better multiplicative approximation factor than $e^n$ for the permanent of a 0-1 matrix?
• RQ2 Is it possible to achieve a polynomial-time approximation for the permanent when the underlying graph is a constant-degree expander?
• RQ3 How can deterministic counting techniques for partial matchings be extended to approximate the permanent?
• RQ4 Can expander decomposition be used to design a deterministic approximation algorithm for general 0-1 matrices?
• RQ5 What is the optimal trade-off between approximation factor and running time in the deterministic setting for permanent computation?

Key findings

• For constant-degree expander graphs, the algorithm achieves a $(1+\epsilon)^n$ multiplicative approximation in polynomial time.
• For general 0-1 matrices, the algorithm achieves the same $(1+\epsilon)^n$ approximation factor in $\exp(O(n^{2/3}\log^3 n))$ time.
• The running time dependence on $\epsilon$ is $\exp(O(\epsilon^{-1/2}))$, which is subexponential in $n$ but not fully polynomial.
• The algorithm improves upon the $e^n$ approximation factor of [LSW00] for expanders, providing a significantly better factor for fixed $\epsilon > 0$.
• The recursive decomposition strategy ensures that the overall approximation factor remains multiplicative across subproblems.
• The analysis relies on bounding binomial coefficients and recursive subproblem sizes, ultimately proving the $\exp(O(n^{2/3}\log^3 n))$ bound on the running time.