Diameter, Eccentricities and Distance Oracle Computations on H-Minor Free Graphs and Graphs of Bounded (Distance) Vapnik–Chervonenkis Dimension

Abstract

Under the strong exponential-time hypothesis, the diameter of general unweighted graphs cannot be computed in truly subquadratic time (in the size of the input), as shown by Roditty and Williams. Nevertheless there are several graph classes for which this can be done such as bounded-treewidth graphs, interval graphs, and planar graphs, to name a few. We propose to study unweighted graphs of constant distance Vapnik–Chervonenkis (VC)-dimension as a broad generalization of many such classes—where the distance VC-dimension of a graph is defined as the VC-dimension of its ball hypergraph whose hyperedges are the balls of all possible radii and centers in . In particular for any fixed , the class of -minor free graphs has distance VC-dimension at most . Our first main result is a Monte Carlo algorithm that on graphs of distance VC-dimension at most , for any fixed , either computes the diameter or concludes that it is larger than in time , where only depends on and the notation suppresses polylogarithmic factors. We thus obtain a truly subquadratic-time parameterized algorithm for computing the diameter on such graphs. Then as a byproduct of our approach, we get a truly subquadratic-time randomized algorithm for constant diameter computation on all the nowhere dense graph classes. The latter classes include all proper minor-closed graph classes, bounded-degree graphs, and graphs of bounded expansion. Before our work, the only known such algorithm was resulting from an application of Courcelle’s theorem; see Grohe, Kreutzer, and Siebertz [J. ACM, 64 (2017), pp. 1–32]. For any graph of constant distance VC-dimension, we further prove the existence of an exact distance oracle in truly subquadratic space, that answers distance queries in truly sublinear time (in the number of vertices). The latter generalizes prior results on proper minor-closed graph classes to a much larger graph class. Finally, we show how to remove the dependency on for any graph class that excludes a fixed graph as a minor. More generally, our techniques apply to any graph with constant distance VC-dimension and polynomial expansion (or equivalently having strongly sublinear balanced separators). As a result for all such graphs one obtains a truly subquadratic-time deterministic algorithm for computing all the eccentricities, and thus both the diameter and the radius. Our approach can be generalized to the -minor free graphs with bounded positive integer weights. We note that all our algorithms for the diameter problem can be adapted for computing the radius, and more generally all the eccentricities. Our approach is based on the work of Chazelle and Welzl who proved the existence of spanning paths with strongly sublinear stabbing number for every hypergraph of constant VC-dimension. We show how to compute such paths efficiently by combining known algorithms for the stabbing number problem with a clever use of -nets, region decomposition, and other partition techniques.