Going Beyond Surfaces in Diameter Approximation

Abstract: Calculating the diameter of an undirected graph requires quadratic running time under the Strong Exponential Time Hypothesis and this barrier works even against any approximation better than 3/2. For planar graphs with positive edge weights, there are known $(1+\varepsilon)$-approximation algorithms with running time $poly(1/\epsilon, \log n) \cdot n$. However, these algorithms rely on shortest path separators and this technique falls short to yield efficient algorithms beyond graphs of bounded genus. In this work we depart from embedding-based arguments and obtain diameter approximations relying on VC set systems and the local treewidth property. We present two orthogonal extensions of the planar case by giving $(1+\varepsilon)$-approximation algorithms with the following running times: 1. $O_h((1/\varepsilon)^{O(h)} \cdot n \log^2 n)$-time algorithm for graphs excluding an apex graph of size h as a minor, 2. $O_d((1/\varepsilon)^{O(d)} \cdot n \log^2 n)$-time algorithm for the class of d-apex graphs. As a stepping stone, we obtain efficient (1+\varepsilon)-approximate distance oracles for graphs excluding an apex graph of size h as a minor. Our oracle has preprocessing time $O_h((1/\varepsilon)^8\cdot n \log n \log W)$ and query time $O((1/\varepsilon)^2 * \log n \log W)$, where $W$ is the metric stretch. Such oracles have been so far only known for bounded genus graphs. All our algorithms are deterministic.