A benchmark QUBO problem inspired by digital halftoning based on the human visual system

Quadratic Unconstrained Binary Optimization (QUBO) is a combinatorial optimization to find an optimal binary solution vector that minimizes the energy value defined by a quadratic formula of binary variables in the vector. As many NP problems can be reduced to QUBO problems, considerable research has gone into developing QUBO solvers running on various computing platforms such as quantum devices, ASICs, FPGAs, GPUs, and optical fibers. For evaluating the performance of QUBO solvers, QUBO problems reduced from combinatorial optimization problems such as MaxCut, Traveling Salesman Problem (TSP), and Quadratic Assignment Problem (QAP) are commonly used. This paper presents a new benchmark QUBO problem inspired by digital halftoning, which aims to find a binary image reproducing an input grayscale image. The characteristics of this QUBO problem reduced from digital halftoning are quite unique. Solutions of the QUBO problem can be converted into binary images, from which we can visually evaluate the solution quality. Good approximation solutions for the QUBO problem can be obtained by a simple heuristic algorithm very quickly. However, it is very hard to find an optimal solution. Moreover, quite interestingly, we can create a QUBO problem for digital halftoning with a known optimal solution for a benchmarking purpose. Thus, we can exactly compute the gap between the optimal solution and an approximate solution obtained by a QUBO solver. We have developed an OpenMP-based sparse QUBO solver that can find good approximate solutions for QUBO problems reduced from digital halftoning. We have evaluated the performance of QUBO solvers including the D-Wave Hybrid solver, Gurobi optimizer, Open.Jij, and our sparse QUBO solver. The experimental results show that our OpenMP-based sparse QUBO solver attains the best performance.