二次无约束二元优化的计算复杂性
Computational Complexity of Quadratic Unconstrained Binary Optimization
摘要：
本文研究了函数问题FP^NP范畴下二次无约束二元优化（QUBO）问题的计算复杂性.我们主要讨论了四个子类：（1）当所有系数都是整数时，QUBO是FP^NP-完全的。(2)当每个系数都是一个常数k的整数下界时，QUBO是FP^NP[log]-完全的。(3)当每个系数都是由常数k上界的整数时，QUBO再次是FP^NP[log]-完全的。(4)当系数只能在集合{1，0，-1}中时，QUBO是FP^NP[log]-完全的。最近的结果在量子退火能够有效地解决QUBO问题，我们的研究结果提供了一个明确的量子退火算法和FP^NP复杂性类分类之间的联系。我们还研究了整系数QUBO问题的决策版本的计算复杂性。我们证明了这个问题是DP-完全的。
英文摘要：
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 four sub-classes: (1) When all coefficients are integers QUBO is FP^NP-complete. (2) When every coefficient is an integer lower bounded by a constant k, QUBO is FP^NP[log]-complete. (3) When every coefficient is an integer upper bounded by a constant k, QUBO is again FP^NP[log]-complete. (4) When coefficients can only be in the set {1, 0, -1}, QUBO is 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.
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