  Many computational problems are intractable through classical computing and,
as Moore's law is drawing to a halt, demand for finding alternative methods in
tackling these problems is growing. Here, we realize a liquid light machine for
the NP-hard max-3-cut problem based on a network of synchronized
exciton-polariton condensates. We overcome the binary limitation of the
decision variables in Ising machines using the continuous-phase degrees of
freedom of a coherent network of polariton condensates. The condensate network
dynamical transients provide optically-fast annealing of the XY Hamiltonian. We
apply the Goemans and Williamson random hyperplane technique, discretizing the
XY ground state spin configuration to serve as ternary decision variables for
an approximate optimal solution to the max-3-cut problem. Applications of the
presented coherent network are investigated in image-segmentation tasks and in
circuit design.
