The Maximum Independent Set (MIS) problem on unit-disk graphs is an NP-hard problem that can be naturally encoded on Rydberg atom systems. While existing mapping schemes enable the encoding of arbitrary graphs, current two-dimensional (2D) embedding approaches have an $O\left(n^2\right)$ atom overhead, limiting the size of problems solvable on near-term hardware. In this work, we present a scalable scheme that maps the MIS problem on arbitrary graphs onto programmable three-dimensional (3D) arrays of Rydberg atoms. By utilizing the extra degree of freedom provided by the third dimension, we develop a graph-reduction algorithm that embeds any graph into a 3D unit-disk graph. We show this embedding requires an overhead of $O(min\{m\sqrt{n},n^2\})$ atoms, where $n$ and $m$ denote the number of vertices and edges of the original graph. Furthermore, we prove that our embedding scheme is optimal for both bounded-degree graphs and dense graphs. Benchmarks on 3-regular, Erdős-Rényi, SAT-derived graphs, and standard MIS benchmark sets demonstrate that the required atom count is one to two orders of magnitude lower than state-of-the-art 2D schemes. Our work establishes a practical route toward solving large-scale, classically intractable MIS instances on near-term Rydberg quantum processors.

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• Atom & ion trapping & guiding
• Electric dipole-dipole interactions
• Quantum algorithms
• Rydberg atoms & molecules