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Sharp Linear Bounds and Exponential Separation for Single-Flip Methods in Absolute Value Equations

Year: 2026
Type: preprint
Abstract: Rohn's sign-accord algorithm is a classical single-coordinate sign-flip method for $Ax-|x|=b$, selecting the least-index mismatched coordinate at each step. We construct rational instances on which this rule requires exactly $2^{n-1}-1$ flips from natural initialization, even when both $\|A^{-1}\|_1$ and $\|A^{-1}\|_\infty$ are arbitrarily small. We propose two alternative selection rules, Gauss--Southwell and steepest single flipping, which require only one flip on the same instances. A discrete potential framework with positive polarity factors identifies solutions with global maximizers. Under positive polarity with ratio $κ$ and $r=\|A^{-1}\|_\infty<1$, both rules terminate from any initial signature after at most $\lfloor sκ(1+r)/(1-r)\rfloor$ flips, where $s$ is the number of nonzero solution coordinates. For fixed $η_1<1/2$ and $η_2<1$, both rules have worst-case flip complexity $O(n)$ on the class $\|A^{-1}\|_1\leη_1$ and $\|A^{-1}\|_\infty\leη_2$. An $n-1$ lower bound from natural initialization shows that this linear order is sharp. Under stronger, explicit sign-correction conditions, each rule makes exactly as many flips as there are initially incorrect signs among the nonzero solution components. Thus, each rule makes at most \(n\) flips, and this bound is optimal. A five-dimensional example requiring six greedy flips shows that the bound of $n$ does not extend to the entire common small-norm class. Building on Radons--Rump sign certificates, we further develop three classes of sign-flip algorithms using elimination guidance, cyclic scanning, and randomized $m$-flipping. Under $ρ(|A^{-1}|)<1/2$, we prove at most $n$ flips for elimination guidance, $n^2$ inspections for cyclic scanning, and $O(n^2/m)$ expected batch updates for randomized $m$-flipping. Exact computations and retained paired experiments distinguish flip counts from computational cost.
DOI: 10.48550/arxiv.2609.12763
Open Access status: green
OpenAlex ID: W7212874994
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