We investigate the effect of crossover in the context of parameterized complexity on a well-known fixed-parameter tractable combinatorial optimization problem known as the closest string problem. We prove that a multi-start (\(\mu\)+1) GA solves arbitrary length-n instances of closest string in \(2^{O(d^2 + d \log k)} \cdot t(n)\) steps in expectation. Here, k is the number of strings in the input set, d is the value of the optimal solution, and \(n \le t(n) \le {\text {poly}}(n)\) is the number of iterations allocated to the (\(\mu\)+1) GA before a restart, which can be an arbitrary polynomial in n. This confirms that the multi-start (\(\mu\)+1) GA runs in randomized fixed-parameter tractable (FPT) time with respect to the above parameterization. On the other hand, if the crossover operation is disabled, we show there exist instances that require \(n^{\varOmega (\log (d+k))}\) steps in expectation. The lower bound asserts that crossover is a necessary component in the FPT running time.

我们研究了在参数化复杂度的情况下交叉对众所周知的固定参数可处理组合优化问题（称为最接近字符串问题）的影响。我们证明了一个多起点（\（\亩\） +1）GA解决了任意的长度- ñ接近串的实例中\（2 ^ {O（d ^ 2 + d \日志K）} \ CDOT T（N ）\）符合预期。这里，k是输入集中的字符串数，d是最优解的值，\（n \ le t（n）\ le {\ text {poly}}（n）\）是重新启动前分配给（\（\ mu \） +1）GA的迭代，它可以是n中的任意多项式。这证实了相对于上述参数化，多起点（\（\ mu \） +1）GA在随机化的固定参数可处理（FPT）时间内运行。另一方面，如果禁用了交叉操作，则表明存在预期需要\（n ^ {\ varOmega（\ log（d + k））} \）步骤的实例。下限断言，交叉是FPT运行时间的必要组成部分。