Fixed-Parameter Tractability of Crossover: Steady-State GAs on the Closest String Problem

Abstract

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.