On approximability of the Permanent of PSD matrices
Farzam Ebrahimnejad, Ansh Nagda, Shayan Oveis Gharan
TL;DR
This work resolves two central questions about PSD permanents: how well they can be approximated and how hard this task is. It introduces a deterministic $e^{-(\gamma+0.9999)n}$-approximation algorithm based on an SDP relaxation and a refined rounding scheme, substantially improving the previous $e^{-(\gamma+1)n}$ bound. On the hardness side, it establishes the first exponential-in-$n$ hardness of approximation for per$(A)$ of PSD matrices, showing NP-hardness to achieve factors $e^{-(\gamma-\varepsilon)n}$; the authors achieve this via an extension of $2\to q$-norm hardness to all $-1<q<2$ and an approximation-preserving reduction to PSD permanents. The paper further connects PSD permanents to the maximizing product of linear forms problem and related tasks, explaining implications for quantum-inspired optimization and complexity of matrix norms. Overall, the results substantially advance understanding of PSD permanents, framing both actionable algorithmic avenues and fundamental hardness barriers with broad connections to matrix analysis and Gaussian-based representations.
Abstract
We study the complexity of approximating the permanent of a positive semidefinite matrix $A\in \mathbb{C}^{n\times n}$. 1. We design a new approximation algorithm for $\mathrm{per}(A)$ with approximation ratio $e^{(0.9999 + γ)n}$, exponentially improving upon the current best bound of $e^{(1+γ-o(1))n}$ [AGOS17,YP22]. Here, $γ\approx 0.577$ is Euler's constant. 2. We prove that it is NP-hard to approximate $\mathrm{per}(A)$ within a factor $e^{(γ-ε)n}$ for any $ε>0$. This is the first exponential hardness of approximation for this problem. Along the way, we prove optimal hardness of approximation results for the $\|\cdot\|_{2\to q}$ ``norm'' problem of a matrix for all $-1 < q < 2$.