Improved approximation of layout problems on random graphs

TL;DR

This paper demonstrates that several classic graph layout problems can be approximated nearly optimally with high probability on Erdős-Rényi random graphs, extending results to directed acyclic graphs.

Contribution

It improves approximation guarantees for layout problems on random graphs and extends these results to directed acyclic graphs, under certain sparsity conditions.

Findings

Approximation factor arbitrarily close to 1 for Erdős-Rényi graphs

High probability guarantees under sparsity conditions

Extension to directed acyclic graphs

Abstract

Inspired by previous work of Diaz, Petit, Serna, and Trevisan (Approximating layout problems on random graphs Discrete Mathematics, 235, 2001, 245--253), we show that several well-known graph layout problems are approximable to within a factor arbitrarily close to 1 of the optimal with high probability for random graphs drawn from an Erd\"os-Renyi distribution with appropriate sparsity conditions. Moreover, we show that the same results hold for the analogous problems on directed acyclic graphs.