P-dit Probabilistic Ising Machine for Solving the Quadratic Assignment Problem

Abstract

Combinatorial optimization problems represent a wide range of real-world scenarios where complicated interactions make it difficult to find the best solution. One example is the quadratic assignment problem (QAP), which involves determining the optimal placement of facilities at set locations which minimizes the products of material flow and facility distance. This representation is descriptive of many real-world scenarios, including the aggregate transportation costs of a supply chain. In this work, a probabilistic Ising machine (PIM) approach is implemented using probabilistic d-dimensional variables (p-dits), which are generalized, multi-state and multi-dimensional extensions to probabilistic bits (p-bits). Each p-dit corresponds to a location and stochastically oscillates between facility assignments based on the influence of the other p-dits. We show that with the same runtime and CPU, the PIM finds the best-known solution on 95% of considered instances from the QAP Library dataset, compared to just 36% for the standard Gurobi solver. For the unique largest problem in the library, a 2 to 3 order-of-magnitude decrease is observed in the time needed to reach specific solution qualities. We also show parallelization of our PIM through GPU implementations. A comparison to state-of-the-art QAP solver algorithms shows that they are consistently outperformed by both CPU and GPU implementations of the p-dit Ising machine.