Highway Dimension: a Metric View

TL;DR

The work introduces a relaxed highway dimension that applies to general metric spaces by hitting approximate shortest paths within balls, enabling algorithmic tools previously restricted to graphs. It proves a PTAS for Subset TSP in spaces with small highway dimension and develops a comprehensive metric toolkit comprising padded decompositions, sparse covers, and tree covers that unlocks numerous applications. The toolkit yields embeddings into l_p spaces, distance oracles, and various approximation schemes for extension and flow problems, highlighting the practical impact for realistic metric spaces such as road networks and doubling spaces. Overall, the paper provides a unified framework linking realistic distance structures to efficient approximation algorithms and data-structural tools.

Abstract

Realistic metric spaces (such as road/transportation networks) tend to be much more algorithmically tractable than general metrics. In an attempt to formalize this intuition, Abraham et al. (SODA 2010, JACM 2016) introduced the notion of highway dimension. A weighted graph $G$ has highway dimension $h$ if for every ball $B$ of radius $\approx 4r$ there is a hitting set of size $h$ hitting all the shortest paths of length $>r$ in $B$. Unfortunately, this definition fails to incorporate some very natural metric spaces such as the grid graph, and the Euclidean plane. We relax the definition of highway dimension by demanding to hit only approximate shortest paths. In addition to generalizing the original definition, this new definition also incorporates all doubling spaces (in particular the grid graph and the Euclidean plane). We then construct a PTAS for TSP under this new definition (improving a QPTAS w.r.t. the original more restrictive definition of Feldmann et al. (SICOMP 2018)). Finally, we develop a basic metric toolkit for spaces with small highway dimension by constructing padded decompositions, sparse covers/partitions, and tree covers. An abundance of applications follow.

Figures (5)

• Figure 1: An example of a graph $G=(V,E,w)$ with small highway dimension, while it's respective shortest path metric has large highway dimension. Fix $\varepsilon>0$. The graph $G$ in the figure consist of a center vertex $s$, which is connected to $n$ pairs of vertices $v_1,u_1,v_2,u_2,\dots,v_n,u_n$ by edges of weight $1$. In addition, each such pair $v_i,u_i$ is connected to a vertex $z_i$ with edges of weight $\alpha=\frac{1}{7+16\varepsilon}$. The graph $G$, for $\varepsilon$ has highway dimension $h(\varepsilon)=1$. Indeed, fix $r=\frac{1}{4+8\varepsilon}$, and consider the ball $B_v=B_G(s,(4+8\varepsilon)\cdot r)=B_G(s,1)$. The induced graph $G[B_v]$ is simply a star graph with $2n$ leaves, all at pairwise distances $2$. In particular $H_\varepsilon=\{s\}$ hits all the (exact) shortest paths. Next fix $r'=\frac{1+\alpha}{4+8\varepsilon}=\frac{1+\frac{1}{7+16\varepsilon}}{4+8\varepsilon}=\frac{2}{7+16\varepsilon}=2\alpha$, and consider the ball $B'_v=B_G(s,(4+8\varepsilon)\cdot r')=B_G(s,1+\alpha)$. Note that the distance from $v_i$ to $u_i$ is $2\alpha=r'$. It follows that $H_\varepsilon=\{s\}$ is still a valid hitting set, as it hits all the shortest paths of length strictly greater than $r$. The other balls have small hitting sets as well. Next consider the shortest path metric $(X=V,d_G)$ of $G$, and the ball $B_v=B_G(s,(4+8\varepsilon)\cdot r)=B_X(s,1)$. For every pair $v_i,u_i$, the distance is $d_X(v_i,u_i)=2\alpha=\frac{2}{7+16\varepsilon}>\frac{1}{4+8\varepsilon}=r$, and thus $H_\varepsilon$ must contain a point from $\{u_i,v_i\}$. It follows that the highway dimension is at least $h(\varepsilon)\ge n$. It is also interesting to note that the doubling dimension of the shortest path metric is $\Theta(\log n)$.
• Figure 2: Summary of new and previous results on the metric toolkit.
• Figure 3: Illustration of the proof of \ref{['lem:hub_bound']}. On the left is illustrated a set of hubs $W=\left\{ x_{1},\dots,x_{10}\right\}$ at distance $(2+\varepsilon)r$ from the ball $B_{v}=B_{X}(v,(2+4\varepsilon)r)$. Every hub $x\in W$ has a corresponding approximate shortest path $Q_{x}$ of length in $(r,(2+\varepsilon)r]$, which is chosen to minimize the distance $d_{X}(v,Q_{x})$ to the center. $L_{1}=\left\{ x_{7},x_{8},x_{9},x_{10}\right\} \subseteq W$ is the set of hubs whose corresponding path $Q_{x}$ lies inside $B_{1}=B_{X}(v,(4+8\varepsilon)r)$. By \ref{['clm:balls_aux']}, $|L_{1}|\le s\cdot h(\varepsilon)$. On the right is illustrated the set of remaining hubs $W\setminus L_{1}$. For a hub $x\in W\setminus L_{1}$, let $a_{x}$ (colored in red) be the closest point in $B_{v}=B_{X}(v,(2+4\varepsilon)r)$ to $x$. It holds that $d_{X}(x,a_{x})<r$ (as otherwise, by minimality of $d_{X}(v,Q_{x})$, $Q_{x}$ would have been chosen as the path from $a_{x}$ to $x$). Let $b_{x}$ be the farthest point on $Q_{x}$ from $x$ (colored in blue). As $d_{X}(x,b_{x})\le w(Q_{x})\le(2+\varepsilon)r$, it follows that $d_{X}(v,b_{x})<(5+8\varepsilon)r$. Set $\ell_{2}=(\frac{5+8\varepsilon}{4+8\varepsilon})r$ and $B_{2}=B_{X}(v,(4+8\varepsilon)\ell_{2})$. Let $L_{2}=\left\{ x_{1},x_{2},x_{3}\right\} \subseteq W\setminus L_{1}$ be all the hubs $x$ such that the distance between the corresponding endpoints of $Q_{x}$ is at least $\ell_{2}$. By \ref{['clm:balls_aux']}, $|L_{2}|\le s\cdot h(\varepsilon)$. Finally, set $\ell_{3}=(2+4\varepsilon)r-(1+\varepsilon)\ell_{2}$. Using the triangle inequality, for $x\in W\setminus(L_{1}\cup L_{2})$ it holds that $d_{X}(x,a_{x})\ge\ell_{3}$, while the entire path from $x$ to $a_{x}$ lies in the ball $B_{3}=B_{X}(v,(4+8\varepsilon)\ell_{3})$. By \ref{['clm:balls_aux']}, $|W\setminus(L_{1}\cup L_{2})|\le s\cdot h(\varepsilon)$.
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Theorems & Definitions (82)

• Definition 2: Highway dimension
• Theorem 3
• Definition 4
• Lemma 5: Packing Property
• proof
• Definition 6: shortest path cover
• Lemma 6
• proof
• Definition 7: towns and sprawl
• Lemma 8