  One prominent application of near-term quantum computing devices is to solve
combinatorial optimization such as non-deterministic polynomial-time hard
(NP-hard) problems. Here we present experiments with Rydberg atoms to solve one
of the NP-hard problems, the maximum independent set (MIS) of graphs. We
introduce the Rydberg quantum wire scheme with auxiliary atoms to engineer
long-ranged networks of qubit atoms. Three-dimensional (3D) Rydberg-atom arrays
are constructed, overcoming the intrinsic limitations of two-dimensional
arrays. We demonstrate Kuratowski subgraphs and a six-degree graph, which are
the essentials of non-planar and high-degree graphs. Their MIS solutions are
obtained by realizing a programmable quantum simulator with the quantum-wired
3D arrays. Our construction provides a way to engineer many-body entanglement,
taking a step toward quantum advantages in combinatorial optimization.
