(1+1) Evolutionary Algorithm on Random Planted Vertex Cover Problems

Abstract

Evolutionary Algorithms are powerful optimization tools that use the power of randomness and inspiration from biology to achieve results. A common combinatorial optimization problem is the recovery of a minimum vertex cover on some graph 𝐺 = (𝑉, 𝐸). In this work, an evolutionary algorithm will be employed on specific instances of the minimum vertex cover problem containing a random planted solution. This situation is common in data networks and translates to a core set of nodes and larger fringe set that are connected to the core. This study introduces a parameterized analysis of a standard (1+1) Evolutionary Algorithm applied to the random planted distribution of vertex cover problems. When the planted cover is at most logarithmic, restarting the (1+1) EA every 𝑂(𝑛 log 𝑛) steps will, within polynomial time, yield a cover at least as small as the planted cover for sufficiently dense random graphs (𝑝 > 0.71). For superlogarithmic planted covers, the (1+1) EA is proven to find a solution within fixed-parameter tractable time in expectation. To complement these theoretical investigations, a series of computational experiments were conducted, highlighting the intricate interplay between planted cover size, graph density, and runtime. A critical range of edge probability was also investigated.