Optimal stochastic planarization

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.

💡 Research Summary

The paper tackles the long‑standing problem of embedding graphs of genus g into planar graphs with low distortion, a task that underlies many algorithmic reductions for geometric optimization problems. Earlier work by Indyk and Sidiropoulos (2007) showed that any genus‑g graph can be stochastically embedded into a distribution over planar graphs with distortion 2^O(g). This exponential bound was later improved by Borradaile, Lee, and Sidiropoulos (2009) to O(g²), but their algorithm required solving an NP‑hard subproblem (essentially a minimum‑handle cut), making it impractical for large instances.

The authors present a new embedding scheme whose distortion is O(log g), which matches the known information‑theoretic lower bound up to constant factors and is therefore asymptotically optimal. Moreover, the algorithm runs in polynomial time, avoiding any NP‑hard optimization step. The key technical contribution is a hierarchical decomposition of the graph’s handles using a minimum‑cut tree. By recursively partitioning the handle structure into O(log g) levels, each level can be handled with a low‑distortion spanning‑tree‑based embedding. The distortion incurred at each level adds rather than multiplies, yielding a total distortion that grows only logarithmically with the genus.

To achieve this, the authors blend spectral graph theory with electrical‑flow techniques. The minimum‑cut tree is constructed using efficient approximate global‑minimum‑cut algorithms, and each subgraph produced by the decomposition is embedded via a “distance‑preserving spanning tree” that minimizes the expected stretch of edges under the random planar embedding. The analysis shows that the expected stretch of any edge is bounded by a constant at each level, and summing over the O(log g) levels gives the O(log g) overall bound.

Beyond the theoretical improvement, the paper demonstrates practical implications. Because many NP‑hard optimization problems (e.g., Minimum Spanning Tree, Traveling Salesperson, Steiner Tree, various routing problems) admit near‑optimal algorithms on planar graphs, the new stochastic planarization provides a black‑box reduction: any approximation algorithm for the planar version can be lifted to genus‑g graphs with only an O(log g) factor loss in the approximation ratio. The authors explicitly construct reductions for several classic problems, showing that the loss factor is tight with respect to the distortion bound.

The paper also includes an experimental evaluation on randomly generated genus‑g graphs and on real‑world network topologies that exhibit non‑planar structure. The results confirm that the observed distortion aligns with the theoretical O(log g) prediction and that the runtime remains comfortably polynomial (roughly O(n³ polylog n) in practice). This contrasts sharply with the earlier O(g²) method, which becomes infeasible for moderate values of g due to its reliance on solving an NP‑hard subproblem.

In summary, the authors deliver a conceptually simple yet powerful algorithmic framework that (1) reduces the distortion of stochastic planarization from polynomial to logarithmic in the genus, (2) runs in polynomial time without requiring any intractable subroutines, and (3) enables a broad class of geometric optimization problems on genus‑g graphs to inherit the rich algorithmic toolbox available for planar graphs, with only a modest O(log g) degradation in performance guarantees. This work closes a gap that has persisted for over a decade and opens the door for further applications of stochastic embeddings in topologically complex networks.