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 Erdös-Renyi distribution with appropriate sparsity conditions. Moreover, we show that the same results hold for the analogous problems on directed acyclic graphs.