Crossover can simulate bounded tree search on a fixed-parameter tractable optimization 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 (μ+1) solves arbitrary length-n instances of closest string in 2O(d2+d log k). poly(n) steps in expectation. Here, k is the number of strings in the input set, and d is the value of the optimal solution. This confirms that the multi-start (μ+1) 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 nΩ(log(d+k) steps in expectation. The lower bound asserts that crossover is a necessary component in the FPT running time.