Analysis of an Exhaustive Search Algorithm in Random Graphs and the $n^{c\log n}$-Asymptotics

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.MSC 2000 Subject Classifications: Primary 05C80, 05C85; secondary 65Q30.

An analytic approach to the asymptotic variance of trie statistics and related structures

“…Many asymptotic patterns such as (2), which most of us take for granted today, were far from being clear in the 1980's, notably in engineering contexts. For example, the minute periodic fluctuations when log p/ log q is rational are often invisible in numerical calculations, leading possibly to wrong conclusions.…”

“…Theorem 6.1. The variance of the internal path length of random tries satisfies V(X n ) n = F 0,2 (r log 1/p n) (log n) 2 h 2 + +F [2] 0,2 (r log 1/p n) log n h + F [3] 0,2 (r log 1/p n) + o(1), and the covariance of N n and X n satisfies Cov(N n , X n ) n = F 0,2 (r log 1/p n) log n h + F ·,· 's are either constants when log p log q ∈ Q or periodic functions with computable Fourier series when log p log q ∈ Q. For simplicity, we give only the expressions in the symmetric case…”

“…In general, polynomial growth rate for g(z) for large |z| implies the same for f in a small sector containing the real axis. The only exception is the functional-differential equation [2] f (z) = f (qz) + g(z), for which the growth is of order z c log z when g grows polynomially for large |z|. Note that this equation is a special case of the so-called "pantograph equations"; see [2] for more information.…”

“…Random models. Random graphs: [2,46,100]; Geometric IID RVs (or order statistics): [18,45,74]; Cantor distributions: [10,48]; Evolutionary trees: [1,75]; Diffusion limited aggregates: [6,78]; Generalized Eden model on trees: [13].…”

Average-case complexity of a branch-and-bound algorithm for maximum independent set, under the $\mathcal{G}(n,p)$ random model

“…Close to the problematic of this paper is the one by [1], where a very nice analysis of the exhaustive search for max independent set is done, and phase transitions between exponential, subexponential, and polynomial average-case complexities under the G(n, p) model are given. The main result there, is an asymptotic upper bound on the average running time (denoted by µ n ) of the exhaustive search when applied on G(n, p) random graphs (formula (1.4) in [1]). It is shown that when p ≫ log 2 n /n, µ n becomes subexponential.…”