  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.
