Embracing the Giant Component

Abstract

Consider a game in which edges of a graph are provided a pair at a time, and the player selects one edge from each pair, attempting to construct a graph with a component as large as possible. This game is in the spirit of recent papers on avoiding a giant component, but here we embrace it.

We analyze this game in the offline and online setting, for arbitrary and random instances, which provides for interesting comparisons. For arbitrary instances, we find a large lower bound on the competitive ratio. For some random instances we find a similar lower bound holds with high probability (whp). If the instance has $\frac{1}{4}(1+\epsilon)n$ random edge pairs, when 0<ε≤ 0.003 then any online algorithm generates a component of size O((logn)^3/2)whp, while the optimal offline solution contains a component of size Ω(n) whp. For other random instances we find the average-case competitive ratio is much better than the worst-case bound. If the instance has $\frac{1}{2}(1-\epsilon)n$ random edge pairs, with 0<ε≤ 0.015, we give an online algorithm which finds a component of size Ω(n) whp.