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.

中文翻译：

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二次无约束二元优化的计算复杂度

在本文中，我们研究了函数问题 FP^NP 分类下的二次无约束二元优化 (QUBO) 问题的计算复杂度。我们关注三个子类：（1）当所有系数都是整数时，QUBO 是 FP^NP 完全的。当每个系数都是一个以常数 k 为下界的整数时，QUBO 是 FP^NP[log]-complete。(3) 当系数只能在集合{1,0,-1}中时，QUBO又是FP^NP[log]-完全的。随着量子退火的最新结果能够有效地解决 QUBO 问题，我们的结果提供了量子退火算法和 FP^NP 复杂性类别分类之间的明确联系。我们还研究了具有整数系数的 QUBO 问题决策版本的计算复杂性。我们证明这个问题是DP完全的。