Fixed-Parameter Tractability of the (1+1) Evolutionary Algorithm on Random Planted Vertex Covers

TL;DR

This paper analyzes the fixed-parameter tractability of the (1+1) Evolutionary Algorithm on random planted vertex cover problems, showing polynomial and FPT results depending on the size of the planted cover and graph density.

Contribution

It provides the first parameterized analysis of the (1+1) EA on planted vertex cover problems, establishing polynomial and fixed-parameter tractable runtimes based on cover size.

Findings

For small (logarithmic) planted covers, the EA finds solutions in polynomial time.

For larger (superlogarithmic) planted covers, the EA operates in fixed-parameter tractable time.

Experimental results illustrate the effects of cover size and graph density on runtime.

Abstract

We present the first parameterized analysis of a standard (1+1) Evolutionary Algorithm on a distribution of vertex cover problems. We show that if the planted cover is at most logarithmic, restarting the (1+1) EA every $O(n \log n)$ steps will find a cover at least as small as the planted cover in polynomial time for sufficiently dense random graphs $p > 0.71$. For superlogarithmic planted covers, we prove that the (1+1) EA finds a solution in fixed-parameter tractable time in expectation. We complement these theoretical investigations with a number of computational experiments that highlight the interplay between planted cover size, graph density and runtime.