Discrete Potential Optimization for Absolute Value Equations: A Sign-Flip Framework with Polynomial Complexity
Solving the AVE $Ax - |x| = b$ is generally NP-hard. Existing approaches mostly operate in continuous variable spaces, with the notable exception of Rohn's sign-accord algorithm, which incurs an exponential worst-case bound of $2^n$ iterations. To overcome this bottleneck, we develop a discrete potential optimization (DPO) framework over the sign-vector set $\{-1, 1\}^n$. Under the $1$-norm condition $\|A^{-1}\|_1 < 1/2$, we establish that the AVE solution corresponds exactly to the global maximizer of this discrete potential function. When $\|A^{-1}\|_1$ is uniformly upper bounded by $1/2$, for rational inputs with maximum magnitude $L$, we develop a unified polynomial-time framework for sign-flip algorithms, which excludes Rohn's sign-accord algorithm. Within this framework, specific single-flip mechanisms (including our new steepest and Gauss-Southwell rules) terminate in $\mathcal{O}(n^2\log(nL))$ iterations, while full-flip updates (equivalent to the classical GNM) require only $\mathcal{O}(n\log(nL))$ iterations. We further relax the assumption by requiring that the spectral radius $ρ(|A^{-1}|)$ be uniformly upper bounded by $1/2$ via rational diagonal scaling. Moreover, a uniformly randomized $m$-flip approach is proven to achieve an expected iteration bound of $\mathcal{O}\bigl(n^3 \log(nL)/m\bigr)$ without requiring explicit diagonal preconditioning. As a corollary, GNM solves the AVE in $\mathcal{O}(n^2 \log(nL))$ iterations, improving on the prior result of finite termination under the stricter condition that $ρ(|A^{-1}|)$ is less than~$1/3$. Crucially, by equivalently reformulating linear complementarity problems (LCPs) as AVEs, we extend this DPO framework to yield GNM and pivot-type methods with polynomial iteration complexity for LCPs. Numerical experiments validate the practical efficiency of GNM and the structural robustness of the proposed sign-flip approaches.