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

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 (μ+1) GA solves arbitrary length-n instances of closest string in 2O(d2+dlogk)·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≤ t(n) ≤ poly (n) is the number of iterations allocated to the (μ+1) GA before a restart, which can be an arbitrary polynomial in n. This confirms that the multi-start (μ+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^Ω(log(d+k)) steps in expectation. The lower bound asserts that crossover is a necessary component in the FPT running time.