Computational Complexity of Quadratic Unconstrained Binary Optimization

09/21/2021

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by   Hirotoshi Yasuoka, et al.

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In this paper, we study the computational complexity of the quadratic unconstrained binary optimization (QUBO) problem under the functional problem FP^NP categorization. We focus on three sub-classes: (1) When all coefficients are integers QUBO is FP^NP-complete. When every coefficient is an integer lower bounded by a constant k, QUBO is FP^NP[log]-complete. (3) When coefficients can only be in the set 1, 0, -1, QUBO is again FP^NP[log]-complete. With recent results in quantum annealing able to solve QUBO problems efficiently, our results provide a clear connection between quantum annealing algorithms and the FP^NP complexity class categorization. We also study the computational complexity of the decision version of the QUBO problem with integer coefficients. We prove that this problem is DP-complete.