A $$(1+{\varepsilon })$$ ( 1 + ε ) -Embedding of Low Highway Dimension Graphs into Bounded Treewidth Graphs

Abstract

Graphs with bounded highway dimension were introduced by Abraham et al. [SODA 2010] as a model of transportation networks. We show that any such graph can be embedded into a distribution over bounded treewidth graphs with arbitrarily small distortion. More concretely, given a weighted graph G = (V, E) of constant highway dimension, we show how to randomly compute a weighted graph H = (V, E ) that distorts shortest path distances of G by at most a 1 + ε factor in expectation, and whose treewidth is polylogarithmic in the aspect ratio of G. Our probabilistic embedding implies quasi-polynomial time approximation schemes for a number of optimization problems that naturally arise in transportation networks, including Travelling Salesman, Steiner Tree, and Facility Location.To construct our embedding for low highway dimension graphs we extend Talwar's [STOC 2004] embedding of low doubling dimension metrics into bounded treewidth graphs, which generalizes known results for Euclidean metrics. We add several non-trivial ingredients to Talwar's techniques, and in particular thoroughly analyse the structure of low highway dimension graphs. Thus we demonstrate that the geometric toolkit used for Euclidean metrics extends beyond the class of low doubling metrics.following formal definition, if dist(u, v) denotes the shortest-path distance between vertices u and v, let B r (v) = {u ∈ V |dist(u, v) ≤ r} be the ball of radius r centred at v. We will also say that a path P lies inside B r (v) if all its vertices lie inside B r (v).Definition 1.1. The highway dimension of a graph G is the smallest integer k such that, for some universal constant c ≥ 4, for every r ∈ R + , and every ball B cr (v) of radius cr, there are at most k vertices in B cr (v) hitting all shortest paths of length more than r that lie in B cr (v).Rather than working with the above definition directly, we often consider the closely related notion of shortest path covers (also introduced in [1]).Definition 1.2. For a graph G and r ∈ R + , a shortest path cover spc(r) ⊆ V is a set of hubs that intersect all shortest paths of length in (r, cr/2] of G. Such a cover is called locally s-sparse for scale r, if no ball of radius cr/2 contains more than s vertices from spc(r).

Travelling on Graphs with Small Highway Dimension

“…Our main result concerns graphs of the smallest possible highway dimension, and shows that for these fully polynomial time approximation schemes (FPTASs) exist, i.e., a (1 + ε)-approximation can be computed in time polynomial in both the input size and 1/ε. Thus at least for this restricted case we obtain a significant improvement over the previously known QPTAS [Fel+18].…”

“…It was in fact left as an open problem in [Fel+18] to determine the hardness of STP and also TSP on graphs of constant highway dimension. Theorem 3 settles this question for STP.…”

“…As suggested in [Fel+18], one may hope to prove that the treewidth of any graph of highway dimension h is, say, O(h polylog(α)). As argued in Section 6, it unfortunately is unlikely that such a bound is generally possible.…”

“…Although these are fairly standard techniques for metrics (cf. [Fel+18]), in our case we need to take special care, since we need to bound the treewidth of the graphs resulting from this reduction, which the standard techniques do not guarantee.…”

“…The main question posed in this paper is whether the structure of graphs with low highway dimension admits PTASs for problems such as TSP and STP, similar to Euclidean or planar instances. It was shown that quasi-polynomial time approximation schemes (QPTASs) exist for these problems [Fel+18], i.e., (1 + ε)-approximation algorithms with runtime 2 polylog(n) assuming that ε and the highway dimension of the input graph are constants. However it was left open whether this can be improved to polynomial time.…”