Layout of random circulant graphs

TL;DR

This paper studies the structure of random circulant graphs and introduces a polynomial-time algorithm to approximate the minimum linear arrangement problem, providing bounds on the approximation error with high probability.

Contribution

It presents a novel polynomial-time approximation algorithm for the minimum linear arrangement problem specifically tailored for random circulant graphs, along with probabilistic error bounds.

Findings

The algorithm efficiently approximates the minimum linear arrangement.

Error bounds hold with high probability for the approximation.

The approach is applicable to large random circulant graphs.

Abstract

A circulant graph H is defined on the set of vertices V=\left\{ 1,\ldots,n\right\} and edges E=\left\{ \left(i,j\right):\left|i-j\right|\equiv s\left(\textrm{mod}n\right),s\in S\right\} , where S\subseteq\left\{ 1,\ldots,\lceil\frac{n-1}{2}\rceil\right\} . A random circulant graph results from deleting edges of H with probability 1-p. We provide a polynomial time algorithm that approximates the solution to the minimum linear arrangement problem for random circulant graphs. We then bound the error of the approximation with high probability.