As an important part of genetic algorithms (GAs), mutation operators is widely used in evolutionary algorithms to solve $\mathcal{NP}$-hard problems because it can increase the population diversity of individual. Due to limitations in mathematical tools, the mutation probability of the mutation operator is primarily empirically set in practical applications. In this paper, we propose a novel reduction method for the 0-1 knapsack problem(0-1 KP) and an improved mutation operator (IMO) based on the assumption $\mathcal{NP}\neq\mathcal{P}$, along with the utilization of linear relaxation techniques and a recent result by Dey et al. (Math. Prog., pp 569-587, 2022). We employ this method to calculate an upper bound of the mutation probability in general instances of the 0-1 KP, and construct an instance where the mutation probability does not tend towards 0 as the problem size increases. Finally, we prove that the probability of the IMO hitting the optimal solution within only a single iteration in large-scale instances is superior to that of the traditional mutation operator.

翻译：作为遗传算法（GAs）的重要组成部分，突变算子因能增加个体种群多样性而被广泛用于求解$\mathcal{NP}$-困难问题的进化算法中。受数学工具限制，实际应用中突变算子的突变概率主要凭经验设定。本文提出了一种新颖的0-1背包问题（0-1 KP）归约方法，并基于$\mathcal{NP}\neq\mathcal{P}$假设，结合线性松弛技术与Dey等人（《数学规划》，第569-587页，2022年）的最新成果，改进了突变算子（IMO）。利用该方法，我们计算了0-1 KP一般实例中突变概率的上界，并构建了一个突变概率不会随问题规模增大而趋近于零的实例。最后证明，在大规模实例中，IMO仅通过单次迭代即可命中最优解的概率优于传统突变算子。