Not all computing problems are created equal. The inherent complexity of processing certain classes of problems using digital computers has inspired the exploration of alternate computing paradigms. Coupled oscillators exhibiting rich spatio-temporal dynamics have been proposed for solving hard optimization problems. However, the physical implementation of such systems has been constrained to small prototypes. Consequently, the computational properties of this paradigm remain inadequately explored. Here, we demonstrate an integrated circuit of thirty oscillators with highly reconfigurable coupling to compute optimal/near-optimal solutions to the archetypally hard Maximum Independent Set problem with over 90% accuracy. This platform uniquely enables us to characterize the dynamical and computational properties of this hardware approach. We show that the Maximum Independent Set is more challenging to compute in sparser graphs than in denser ones. Finally, using simulations we evaluate the scalability of the proposed approach. Our work marks an important step towards enabling application-specific analog computing platforms to solve computationally hard problems.

In this work, we experimentally demonstrate an integrated circuit (IC) of 30 relaxation oscillators with reconfigurable capacitive coupling to solve the NP-Hard Maximum Cut (Max-Cut) problem. We show that under the influence of an external second-harmonic injection signal, the oscillator phases exhibit a bi-partition which can be used to calculate a high quality approximate Max-Cut solution. Leveraging the all-to-all reconfigurable coupling architecture, we experimentally evaluate the computational properties of the oscillators using randomly generated graph instances of varying size and edge density (ƞ). Further, comparing the Max-Cut solutions with the optimal values, we show that the oscillators (after simple post-processing) produce a Max-Cut that is within 99% of the optimal value in 28 of the 36 measured graphs; importantly, the oscillators are particularly effective in dense graphs with the Max-Cut being optimal in 7 out of 9 measured graphs with ƞ=0.8. Our work marks a step towards creating an efficient, room-temperature-compatible non-Boolean hardwarebased solver for hard combinatorial optimization problems.

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Graph coloring is a NP-hard problem, and computing the solution on a digital computer entails an exponential increase in the computing resources (time, memory) with increasing problem size. This has motivated the search for alternate and more efficient non-Boolean approaches. Here, we experimentally demonstrate the solution to this problem using the phase dynamics of coupled oscillators. Using a 30-oscillator IC platform with reconfigurable all-to-all coupling and minimal post-processing, our approach achieves 98% accuracy in detecting (near-) optimal solutions within 1 color of the optimal solution in comparison to the 77% accuracy achieved with the heuristic Johnson algorithm. Additionally, we propose a new local search-based post-processing scheme to improve the quality of the coloring solution. Finally, using circuit simulations, we demonstrate the scalability and speed up (~ 100×) achievable with the above approach in larger graphs.

The equivalence between the natural minimization of energy in a dynamical system and the minimization of an objective function characterizing a combinatorial optimization problem offers a promising approach to designing dynamical system inspired computational models and solvers for such problems. For instance, the ground state energy of coupled electronic oscillators, under second harmonic injection, can be directly mapped to the optimal solution of the Maximum Cut problem. However, prior work has focused on a limited set of such problems. Therefore, in this work, we formulate computing models based on synchronized oscillator dynamics for a broad spectrum of combinatorial optimization problems ranging from the Max-K-Cut (the general version of the Maximum Cut problem) to the Traveling Salesman Problem. We show that synchronized oscillator dynamics can be engineered to solve these different combinatorial optimization problems by appropriately designing the coupling function and the external injection to the oscillators. Our work marks a step forward towards expanding the functionalities of oscillator-based analog accelerators, and furthers the scope of dynamical system solvers for combinatorial optimization problems.