Algorithmic Thresholds for Refuting Random Polynomial Systems

TL;DR

This paper investigates the minimum number of polynomial equations needed for efficient algorithms to certify unsatisfiability in random polynomial systems, using sum-of-squares relaxations and lower bounds.

Contribution

It establishes nearly tight bounds on the algorithmic threshold for refuting random polynomial systems via sum-of-squares methods and polynomial lower bounds.

Findings

Sum-of-squares relaxations refute systems when m ≥ O(n)·(n/d)^{D-1}.

Lower bounds suggest the trade-off between relaxation degree and equations count is nearly optimal.

Results imply an algorithmic threshold at m ≈ n^{D-1} for subexponential algorithms.

Abstract

Consider a system of $m$ polynomial equations $\{p_i(x) = b_i\}_{i \leq m}$ of degree $D\geq 2$ in $n$-dimensional variable $x \in \mathbb{R}^n$ such that each coefficient of every $p_i$ and $b_i$s are chosen at random and independently from some continuous distribution. We study the basic question of determining the smallest $m$ -- the algorithmic threshold -- for which efficient algorithms can find refutations (i.e. certificates of unsatisfiability) for such systems. This setting generalizes problems such as refuting random SAT instances, low-rank matrix sensing and certifying pseudo-randomness of Goldreich's candidate generators and generalizations. We show that for every $d \in \mathbb{N}$, the $(n+m)^{O(d)}$-time canonical sum-of-squares (SoS) relaxation refutes such a system with high probability whenever $m \geq O(n) \cdot (\frac{n}{d})^{D-1}$. We prove a lower bound in the restricted low-degree polynomial model of computation which suggests that this trade-off between SoS degree and the number of equations is nearly tight for all $d$. We also confirm the predictions of this lower bound in a limited setting by showing a lower bound on the canonical degree-$4$ sum-of-squares relaxation for refuting random quadratic polynomials. Together, our results provide evidence for an algorithmic threshold for the problem at $m \gtrsim \widetilde{O}(n) \cdot n^{(1-\delta)(D-1)}$ for $2^{n^{\delta}}$-time algorithms for all $\delta$.