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

TL;DR

This paper investigates the expected computational cost of a naive exhaustive search algorithm for maximum independent sets in random graphs, revealing an unusual asymptotic order and analyzing its distribution.

Contribution

It introduces a detailed analysis of the n^{c log n} asymptotic behavior of the algorithm's cost and demonstrates the normal distribution of this cost under a random graph model.

Findings

Expected cost is of order n^{c log n} in random graphs.

The distribution of the algorithm's cost is asymptotically normal.

The developed approach can be applied to other graph algorithms.

Abstract

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.