Optimal Stochastic Planarization

It has been shown by Indyk and Sidiropoulos \cite{indyk_genus} that any graph of genus $g>0$ can be stochastically embedded into a distribution over planar graphs with distortion $2^{O(g)}$. This bound was later improved to $O(g^2)$ by Borradaile, Lee and Sidiropoulos \cite{BLS09}. We give an embedding with distortion $O(\log g)$, which is asymptotically optimal. Apart from the improved distortion, another advantage of our embedding is that it can be computed in polynomial time. In contrast, the algorithm of \cite{BLS09} requires solving an NP-hard problem. Our result implies in particular a reduction for a large class of geometric optimization problems from instances on genus-$g$ graphs, to corresponding ones on planar graphs, with a $O(\log g)$ loss factor in the approximation guarantee.