Analysis of an exhaustive search algorithm in random graphs and the n^{clog n} -asymptotics

We analyze the cost used by a naive exhaustive search algorithm for finding a maximum independent set in random graphs under the usual G_{n,p} -model where each possible edge appears independently with the same probability p. The expected cost turns out to be of the less common asymptotic order n^{c\log n}, which we explore from several different perspectives. Also we collect many instances where such an order appears, from algorithmics to analysis, from probability to algebra. The limiting distribution of the cost required by the algorithm under a purely idealized random model is proved to be normal. The approach we develop is of some generality and is amenable for other graph algorithms.

💡 Research Summary

The paper investigates the average running time of the most elementary exhaustive‑search algorithm for finding a maximum independent set (MIS) in a random graph drawn from the Erdős–Rényi Gₙ,ₚ model. The algorithm proceeds by picking a vertex v, branching on whether v belongs to the MIS, and recursively solving the resulting sub‑instances. The total number of recursive calls, denoted Xₙ, satisfies the distributional recurrence
Xₙ ≡ Xₙ₋₁ + X*_{n‑1‑Binom(n‑1,p)} (n > 2),
with X₀ = 0, X₁ = 1, and where the two terms on the right‑hand side are dependent. Consequently, the expected cost μₙ = E