Graph-theoretic Formulation of QUBO for Scalable Local Search on GPUs

Abstract

Heuristic solvers of the Quadratic Unconstrained Binary Optimization (QUBO) problems have been rapidly studied because of their potentialities to solve a wide range of com-binatorial optimization problems in a unified manner. QUBO solvers, however, face a limitation in the numbers of variables. To overcome this limitation, we reformulate QUBO as a maximum weight induced subgraph problem (MISG) and propose a graph-theoretic approach for solving MISG. It is a problem to find an induced subgraph with the maximum total weight for a given edge-weighted undirected graph. Naive local search that updates only one vertex at a time requires many iterations to obtain good solutions for large-scale problems. To accelerate it, we introduce a new technique to update multiple vertices that can improve the solution at the same time. We have implemented a MISG solver based on our algorithm on NVIDIA A100 GPUs, where flip operations are parallelized with GPU multithreaded computing. Our solver is scalable and supports up to 2³⁰(1G) variables. Our results show that our algorithm outperforms the Gurobi optimizer in terms of solution qualities and the execution time.