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 2 O ( d 2 + d log k ) · 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.