Optimal stochastic planarization

TL;DR

This paper presents an optimal stochastic embedding of genus-g graphs into planar graphs with logarithmic distortion, improving previous bounds and enabling efficient computation and better approximation guarantees for geometric optimization problems.

Contribution

It introduces a polynomial-time algorithm for embedding genus-g graphs into planar graphs with O(log g) distortion, improving upon prior bounds and computational complexity.

Findings

Embedding distortion is asymptotically optimal at O(log g)

The embedding can be computed in polynomial time

Reduces geometric optimization problems on genus-g graphs to planar graphs with logarithmic loss

Abstract

It has been shown by Indyk and Sidiropoulos [IS07] 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 [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 [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.