Improved approximation of layout problems on random graphs

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.