Travelling on Graphs with Small Highway Dimension

TL;DR

This paper investigates the complexity and approximation algorithms for TSP and Steiner Tree problems on graphs with low highway dimension, providing an FPTAS for highway dimension 1 and establishing NP-hardness results.

Contribution

It introduces an FPTAS for TSP on graphs with highway dimension 1 and proves NP-hardness for TSP on graphs with highway dimension 6, advancing understanding of these problems in transportation network models.

Findings

FPTAS developed for TSP when highway dimension is 1

NP-hardness of Steiner Tree in low highway dimension graphs

NP-hardness of TSP for highway dimension 6

Abstract

We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP) in graphs of low highway dimension. This graph parameter was introduced by Abraham et al. [SODA 2010] as a model for transportation networks, on which TSP and STP naturally occur for various applications in logistics. It was previously shown [Feldmann et al. ICALP 2015] that these problems admit a quasi-polynomial time approximation scheme (QPTAS) on graphs of constant highway dimension. We demonstrate that a significant improvement is possible in the special case when the highway dimension is 1, for which we present a fully-polynomial time approximation scheme (FPTAS). We also prove that STP is weakly NP-hard for these restricted graphs. For TSP we show NP-hardness for graphs of highway dimension 6, which answers an open problem posed in [Feldmann et al. ICALP 2015].

Equations

$$\text{BODY}$$

Full text

1 Introduction

Two fundamental optimization problems already included in Karp’s initial list of 21 $\mathsf{NP}$-complete problems [Kar72] are the Travelling Salesperson problem (TSP) and the Steiner Tree problem (STP). Given an undirected graph $G=(V,E)$ with non-negative edge weights $w:E\to\mathbb{R}^{+}$, the TSP asks to find the shortest closed walk in $G$ visiting all nodes of $V$. Besides its fundamental role in computational complexity and combinatorial optimization, this problem has a variety of applications ranging from circuit manufacturing [GH91, LR75] and scientific imaging [BS89] to vehicle routing problems [LND85] in transportation networks. For the STP, a subset $R\subseteq V$ of nodes is marked as terminals. The task is to find a weight-minimal connected subgraph of $G$ containing the terminals. It has plenty of fundamental applications in network design including telecommunication networks [Lju+06], computer vision [CG18], circuit design [Hel+11], and computational biology [Cho+13, LAS16], but also lies at the heart of line planning in public transportation [BNP09].

Both TSP and STP are $\mathsf{APX}$-hard in general [CC08, KLS15, Lam14, Aro+92, BP89, PV06] implying that, unless $\mathsf{P}=\mathsf{NP}$, none of these problems admit a polynomial-time approximation scheme (PTAS), i.e., an algorithm that computes a $(1+{\varepsilon})$-approximation in polynomial time for any given constant ${\varepsilon}>0$. On the other hand, for restricted inputs PTASs do exist, e.g., for planar graphs [BKK07, Kle08, GEP95, Aro+98], Euclidean and Manhattan metrics [ARR98, Mit99], and more generally low doubling111A metric is said to have doubling dimension $d$ if for all $r>0$ every ball of radius $r$ can be covered by at most $2^{d}$ balls of half the radius $r/2$. metrics [BGK12].

We study another class of graphs captured by the notion of highway dimension, which was proposed by [Abr+10]. This graph parameter models transportation networks and is thus of particular importance in terms of applications for both TSP and STP. On a high level, the highway dimension is based on the empirical observation of [Bas+07, BFM09] that travelling from a point in a network to a sufficiently distant point on a shortest path always passes through a sparse set of “hubs”. The following formal definition is taken from [Fel18] and follows the lines of [Abr+10].222It is often assumed that all shortest paths are unique when defining the highway dimension, since this allows good polynomial approximations of this graph parameter [Abr+11]. In this work however, we do not rely on these approximations, and thus do not require uniqueness of shortest paths. Here the distance between two vertices is the length of the shortest path between them, according to the edge weights. The ball $B_{v}(r)$ of radius $r$ around a vertex $v$ contains all vertices with distance at most $r$ from $v$.

**Definition **\thedfn

For a scale $r\in\mathbb{R}_{>0}$, let ${\mathcal{P}}_{(r,2r]}$ denote the set of all vertex sets of shortest paths with length in $(r,2r]$. A shortest path cover for scale $r$ is a hitting set for $\mathcal{P}_{(r,2r]}$, i.e., a set $\operatorname{\textsc{spc}}(r)\subseteq V$ such that $|\operatorname{\textsc{spc}}(r)\cap P|\neq\emptyset$ for all $P\in{\mathcal{P}}_{(r,2r]}$. The vertices of $\operatorname{\textsc{spc}}(r)$ are the hubs for scale $r$. A shortest path cover $\operatorname{\textsc{spc}}(r)$ is locally $h$-sparse, if $|\operatorname{\textsc{spc}}(r)\cap B_{v}(2r)|\leq h$ for all vertices $v\in V$. The highway dimension of $G$ is the smallest integer $h$ such that there is a locally $h$-sparse shortest path cover $\operatorname{\textsc{spc}}(r)$ for every scale $r\in\mathbb{R}_{>0}$ in $G$.

The algorithmic consequences of this graph parameter were originally studied in the context of road networks [Abr+16, Abr+10, Abr+11], which are conjectured to have fairly small highway dimension. Road networks are generally non-planar due to overpasses and tunnels, and are also not Euclidean due to different driving or transmission speeds. This is even more pronounced in public transportation networks, where large stations have many incoming connections and plenty of crossing links, making Euclidean (or more generally low doubling) and planar metrics unsuitable as models. Here the highway dimension is better suited, since longer connections are serviced by larger and sparser stations (such as train stations and airports) that can act as hubs.

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., $(1+{\varepsilon})$-approximation algorithms with runtime $2^{\textrm{polylog}(n)}$ assuming that ${\varepsilon}$ and the highway dimension of the input graph are constants. However it was left open whether this can be improved to polynomial time.

1.1 Our results

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 $(1+{\varepsilon})$-approximation can be computed in time polynomial in both the input size and $1/{\varepsilon}$. Thus at least for this restricted case we obtain a significant improvement over the previously known QPTAS [Fel+18].

Theorem 1

Both Travelling Salesperson and Steiner Tree admit an FPTAS on graphs with highway dimension $1$.

From an application point of view, so-called hub-and-spoke networks that can typically be seen in air traffic networks can be argued to have very small highway dimension close to $1$: their star-like structure implies that hubs are needed at the centers of stars only, where all shortest paths converge. From a more theoretical viewpoint, we show that surprisingly the STP problem is non-trivial on graphs highway dimension $1$, since it is still $\mathsf{NP}$-hard even on this very restricted case. Interestingly, together with Theorem 1 this implies [Vaz01] that STP is weakly $\mathsf{NP}$-hard on graphs of highway dimension $1$. This is in contrast to planar graphs or Euclidean metrics, for which the problem is strongly $\mathsf{NP}$-hard.

Theorem 2

The Steiner Tree problem is weakly $\mathsf{NP}$-hard on graphs with highway dimension $1$.

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 2 settles this question for STP. We also answer the question for TSP, but in this case we are not able to bring down the highway dimension to $1$ so that the following theorem does not complement Theorem 1 tightly.

Theorem 3

The Travelling Salesperson problem is $\mathsf{NP}$-hard on graphs with highway dimension $6$.

1.2 Techniques

We present a step towards a better understanding of low highway dimension graphs by giving new structural insights on graphs of highway dimension $1$. It is not hard to find examples of (weighted) complete graphs with highway dimension $1$ (cf. [Fel+18]), and thus such graphs are not minor-closed. Nevertheless, it was suggested in [Fel+18] that the treewidth of low highway dimension graphs might be bounded polylogarithmically in terms of the aspect ratio $\alpha$, which is the maximum distance divided by the minimum distance between any two vertices of the input graph.

**Definition **\thedfn

*A tree decomposition of a graph $G=(V,E)$ is a tree $D$ where each node $v$ is labelled with a bag $X_{v}\subseteq V$ of vertices of $G$, such that the following holds:

(a) $\bigcup_{v\in V(D)}X_{v}=V$,

(b) for every edge $\{u,w\}\in E$ there is a node $v\in V(D)$ such that $X_{v}$ contains both $u$ and $w$, and

(c) for every $v\in V$ the set $\{u\in V(D)\mid v\in X_{u}\}$ induces a connected subtree of $D$.

The width of the tree decomposition is $\max\{|X_{v}|-1\mid v\in V(D)\}$. The treewidth of a graph $G$ is the minimum width among all tree decompositions for $G$.*

As suggested in [Fel+18], one may hope to prove that the treewidth of any graph of highway dimension $h$ is, say, $O(h\,\textrm{polylog}(\alpha))$. As argued in Section 6, it unfortunately is unlikely that such a bound is generally possible. In contrast to this, our main structural insight on graphs of highway dimension $1$ is that they have treewidth $O(\log\alpha)$. This implies FPTASs for TSP and STP, since we may reduce the aspect ratio of any graph with $n$ vertices to $O(n/{\varepsilon})$ and then use algorithms by [Bod+13] to compute optimum solutions to TSP and STP in graphs of treewidth $t$ in $2^{O(t)}n$ time. Since reducing the aspect ratio distorts the solution by a factor of $1+{\varepsilon}$, this results in an approximation scheme. 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.

It remains an intriguing open problem to understand the complexity and structure of graphs of constant highway dimension larger than $1$.

1.3 Related work

The Travelling Salesperson problem (TSP) is among Karp’s initial list of 21 $\mathsf{NP}$-complete problems [Kar72]. For general metric instances, the best known approximation algorithm is due to [Chr76] and computes a solution with cost at most $3/2$ times the LP value. For unweighted instances, the best known approximation guarantee is $7/5$ and is due to [SV14]. In general the problem is $\mathsf{APX}$-hard [KLS15, Lam14, PV06]. For geometric instances where the nodes are points in $\mathbb{R}^{d}$ and distances are given by some $l_{p}$-norm, there exists a PTAS [Aro98, Mit99] for fixed $d$. When $d=\log n$, the problem is $\mathsf{APX}$-hard [Tre00]. [KL06] generalized the PTAS to hyperbolic space. [GEP95] gave a PTAS for unweighted planar graphs which was later generalized by [Aro+98] to the weighted case. For improvements of the running time see [Kle08].

The Steiner Tree problem (STP) is contained in Karp’s list of $\mathsf{NP}$-complete problems as well [Kar72]. The best approximation algorithm for general metric instances is due to [Byr+10] and computes a solution with cost at most $\ln(4)+\epsilon<1.39$ times that of an LP relaxation. Their algorithm improved upon previous results by, e.g., [RZ05] and [HP99]. Also the STP is $\mathsf{APX}$-hard [CC08] in general. For Euclidean distances and nodes in $\mathbb{R}^{d}$ with $d$ constant there is a PTAS due to [Aro98]. For $d=\log|R|/\log\log|R|$ where $R$ is the terminal set, the problem is $\mathsf{APX}$-hard [Tre00]. For planar graphs, there is a PTAS for STP [BKK07], and even for the more general Steiner Forest problem for graphs with bounded genus [BHM11]. Note that STP remains $\mathsf{NP}$-complete for planar graphs [GJ77].

It is worth mentioning that alternate definitions of the highway dimension exist.333See [Fel+18, Section 9] and [Blu19] for detailed discussions on different definitions of the highway dimension. In particular, in a follow-up paper to [Abr+10], [Abr+16] define a version of the highway dimension, which implies that the graphs also have bounded doubling dimension. A related model for transportation networks was given by [KV17] via the so-called skeleton dimension, which also implies bounded doubling dimension. Hence for these definitions, [BGK12] already provide a PTAS for TSP. The highway dimension definition used here (cf. Section 1) on the other hand allows for metrics of large doubling dimension as noted by [Abr+10]: a star has highway dimension $1$ (by using the center vertex to hit all paths), but its doubling dimension is unbounded. While it may be reasonable to assume that road networks (which are the main concern in the works of [Abr+16, Abr+10, Abr+11]) have low doubling dimension, there are metrics modelling transportation networks for which it can be argued that the doubling dimension is large, while the highway dimension should be small. These settings are better captured by Section 1. For instance, the so-called hub-and-spoke networks that can typically be seen in air traffic networks are star-like networks and are unlikely to have small doubling dimension while still having very small highway dimension close to $1$. Thus in these examples it is reasonable to assume that the doubling dimension is a lot larger than the highway dimension.

[Fel+18] showed that graphs with low highway dimension can be embedded into graphs with low treewidth. This embedding gives rise to a QPTAS for both TSP and STP but also other problems. However, the result in [Fel+18] is only valid for a less general definition of the highway dimension from [Abr+11], i.e., there are graphs which have constant highway dimension according to Section 1 but for which the algorithm of [Fel+18] cannot be applied. For the less general definition from [Abr+11], [BKS18] give a PTAS for Bounded-Capacity Vehicle Routing in graphs of bounded highway dimension. Also the $k$-Center problem has been studied on graphs of bounded highway dimension, both for the less general definition [BKS18] and the more general one used here [FM18, Fel18].

2 Structure of graphs with highway dimension 1

In this section, we analyse the structure of graphs with highway dimension $1$. To this end, let us fix a graph $G$ with highway dimension $1$ and a shortest path cover $\operatorname{\textsc{spc}}(r)$ for each scale $r\in\mathbb{R}^{+}$. As a preprocessing, we remove edges that are longer than the shortest path between their endpoints, so that the triangle inequality holds.

We begin by analysing the structure of the graph $G_{\leq 2r}$, which is spanned by all edges of the input graph $G$ of length at most $2r$. If $G$ has highway dimension $1$ it exhibits the following key property.

**Lemma **\thelem

*Let $G$ be a metric graph with highway dimension $1$, $r\in\mathbb{R}^{+}$ a scale, and $\operatorname{\textsc{spc}}(r)$ a shortest path cover for scale $r$. Then, every connected component of $G_{\leq 2r}$ contains at most one hub. *

Proof

For the sake of contradiction, let $r\in\mathbb{R}^{+}$ and let $x,y\in\operatorname{\textsc{spc}}(r)$ be a closest pair of distinct hubs in some component of $G_{\leq 2r}$. Let further $P$ be a shortest path in $G_{\leq 2r}$ between $x$ and $y$ using only edges of length at most $2r$. (Note that $P$ need not be a shortest path between $x$ and $y$ in $G$.) In particular, there is no other hub from $\operatorname{\textsc{spc}}(r)\setminus\{x,y\}$ along $P$. This implies that every edge of $P$ that is not incident to either $x$ or $y$ must be of length at most $r$, since otherwise the edge would be a shortest path of length $(r,2r]$ between its endpoints (using that $G$ is metric) contradicting the fact that $\operatorname{\textsc{spc}}(r)$ is a shortest path cover for scale $r$.

Since the highway dimension of $G$ is $1$, any ball $B_{w}(2r)$ around a vertex $w\in V(P)$ contains at most one of the hubs $x,y\in\operatorname{\textsc{spc}}(r)$. Let $x^{\prime},y^{\prime}\in P$ be the vertices indicent to $x$ and $y$ along $P$, respectively. Since the length of the edge $\{x,x^{\prime}\}$ is at most $2r$, the ball $B_{x^{\prime}}(2r)$ must contain $x$ and, by the observation above, it cannot contain $y$ (in particular $\{x,y\}$ is not an edge). Symmetrically, the ball $B_{y^{\prime}}(2r)$ contains $y$ but not $x$. Consequently, $x^{\prime}\neq y^{\prime}$ and neither of these two vertices can be a hub of scale $r$, i.e., the path $P$ contains at least two vertices different from $x$ and $y$.

Let $V_{x}=\{w\in V:\operatorname{dist}(x,w)<\operatorname{dist}(y,w)\}$ contain all vertices closer to $x$ than to $y$, where $\operatorname{dist}(\cdot,\cdot)$ refers to the distance in the original graph $G$. As all edge weights are strictly positive, we have that $\operatorname{dist}(x,y)>0$ and thus $y\notin V_{x}$. Since $P$ starts with vertex $x\in V_{x}$ and ends with vertex $y\notin V_{x}$ we deduce that there is an edge $\{u,v\}$ of $P$ such that $u\in V_{x}$ and $v\notin V_{x}$. In particular, $\operatorname{dist}(x,u)<\operatorname{dist}(y,u)$ and $\operatorname{dist}(y,v)\leq\operatorname{dist}(x,v)$. We must have $\{u,v\}\neq\{y^{\prime},y\}$, since otherwise $\operatorname{dist}(x,y^{\prime})<\operatorname{dist}(y,y^{\prime})\leq 2r$ and hence $B_{y^{\prime}}(2r)$ would contain $x$. Similarly, we have $\{u,v\}\neq\{x,x^{\prime}\}$, since otherwise $B_{x^{\prime}}(2r)$ would contain $y$. Note that, by definition, $u\neq y$ and $v\neq x$, and hence $x,y\notin\{u,v\}$. Consequently, since every edge of $P$ not incident to either $x$ or $y$ must have length at most $r$, we conclude that $\{u,v\}$ has length at most $r$.

Finally, consider the scale $r^{\prime}\in\mathbb{R}^{+}$, defined such that $2r^{\prime}=\operatorname{dist}(x,u)+\operatorname{dist}(u,v)$. Let $Q$ and $Q^{\prime}$ denote shortest paths between $x,u$ and $v,y$ in $G$, respectively. Then the ball $B_{v}(2r^{\prime})$ around $v$ contains $Q$ by definition of $r^{\prime}$. From $\operatorname{dist}(y,v)\leq\operatorname{dist}(x,v)\leq\operatorname{dist}(x,u)+\operatorname{dist}(u,v)=2r^{\prime}$ it follows that $B_{v}(2r^{\prime})$ contains $Q^{\prime}$ as well. Also, $\operatorname{dist}(y,v)\leq\operatorname{dist}(x,v)$ means that $B_{v}(2r)$ cannot contain $x$, and hence $2r^{\prime}=\operatorname{dist}(x,u)+\operatorname{dist}(u,v)\geq\operatorname{dist}(x,v)>2r$, which implies $r^{\prime}>r$. W.l.o.g., assume that $\operatorname{dist}(x,u)\leq\operatorname{dist}(v,y)$ (otherwise consider scale $2r^{\prime}=\operatorname{dist}(y,v)+\operatorname{dist}(u,v)$ and the ball $B_{u}(2r^{\prime})$). Our earlier observation that $\operatorname{dist}(u,v)\leq r$ with $r<r^{\prime}$ then yields $\operatorname{dist}(v,y)\geq\operatorname{dist}(x,u)=2r^{\prime}-\operatorname{dist}(u,v)>r^{\prime}$. In other words, the lengths of both paths $Q$ and $Q^{\prime}$ are in $(r^{\prime},2r^{\prime}]$, and so they both need to contain a hub of $\operatorname{\textsc{spc}}(r^{\prime})$. However, by definition of $u,v$, the paths $Q$ and $Q^{\prime}$ are vertex disjoint, which means that the ball $B_{v}(2r^{\prime})$, which contains $Q$ and $Q^{\prime}$, also contains at least two hubs from $\operatorname{\textsc{spc}}(r^{\prime})$. This is a contradiction with $G$ having highway dimension $1$. ∎

Given a graph $G$, we now consider graphs $G_{\leq 2r}$ for exponentially growing scales. In particular, for any integer $i\geq 0$ we define the scale $r_{i}=2^{i}$ and call a connected component of $G_{\leq 2r_{i}}$ a level-$i$ component. Note that the level-$i$ components partition the graph $G$, and that the level-$i$ components are a refinement of the level-$(i+1)$ components, i.e., every level-$i$ component is contained in some level-$(i+1)$ component. W.l.o.g., we scale the edge weights of the graph such that $\min_{e\in E}w(e)=3$, so that there are no edges on level [math], and every level-[math] component is a singleton. Let $\alpha=\frac{\max_{u\neq v}\operatorname{dist}(u,v)}{\min_{u\neq v}\operatorname{dist}(u,v)}=\frac{\max_{u\neq v}\operatorname{dist}(u,v)}{3}$ be the aspect ratio of $G$. In our applications we may assume that $G$ is connected, so that there is exactly one level-$(1+\lceil\log_{2}(\alpha)\rceil)$ component containing all of $G$.

Since every edge is a shortest path between its endpoints, every edge $e=\{u,v\}$ that connects a vertex $u$ of a level-$i$ component $C$ with a vertex $v$ outside $C$ is hit by a hub of $\operatorname{\textsc{spc}}(r_{j})$, where $j$ is the level for which $w(e)\in(r_{j},2r_{j}]$. Moreover, since $v$ lies outside $C$, we have $w(e)>2r_{i}$ and, thus, $j\geq i+1$. The following definition captures the set of the hubs through which edges can possibly leave $C$.

**Definition **\thedfn

Let $C$ be a level-$i$ component of $G$. We define the set of interface points of $C$ as $I_{C}:=\bigcup_{j\geq i}\{u\in\operatorname{\textsc{spc}}(r_{j}):\operatorname{dist}_{C}(u)\leq 2r_{j}\},$ where $\operatorname{dist}_{C}(u)$ denotes the minimum distance from $u$ to a vertex in $C$ (if $u\in C$, $\operatorname{dist}_{C}(u)=0$).

Note that, for technical reasons, we explicitly add every hub at level $i$ of a component to its set of interface points as well, even if such a hub does not connect the component with any vertex outside at distance more than $2r_{i}$.

**Lemma **\thelem

If $G$ has highway dimension $1$, then each interface $I_{C}$ of a level-$i$ component $C$ contains at most one hub for each level $j\geq i$.

Proof

Assume that there are two hubs $u,v\in\operatorname{\textsc{spc}}(r_{j})$ in $I_{C}$, and recall that we preprocessed the graph so that the triangle inequality holds. Then $u$ and $v$ must be contained in the same level-$j$ component $C^{\prime}$, since $u$ and $v$ are connected to $C$ with edges of length at most $2r_{j}$ (or are contained in $C$) and $C\subseteq C^{\prime}$. This contradicts Section 2. ∎

Using level-$i$ components and their interface points we can prove that the treewidth of a graph with highway dimension $1$ is bounded in terms of the aspect ratio.

**Lemma **\thelem

If a graph $G$ has highway dimension $1$ and aspect ratio $\alpha$, its treewidth is at most $1+\lceil\log_{2}(\alpha)\rceil$.

Proof

The tree decomposition of $G$ is given by the refinement property of level-$i$ components. That is, let $D$ be a tree that contains a node $v_{C}$ for every level-$i$ component $C$ for all levels $0\leq i\leq 1+\lceil\log_{2}(\alpha)\rceil$. For every node $v_{C}$ we add an edge in $D$ to node $v_{C^{\prime}}$, if $C$ is a level-$i$ component, $C^{\prime}$ is a level-$(i+1)$ component, and $C\subseteq C^{\prime}$. The bag $X_{C}$ for node $v_{C}$ contains the interface points $I_{C}$. For a level-[math] component $C$ the bag $X_{C}$ additionally contains the single vertex $u$ contained in $C$.

Clearly, the tree decomposition has Property (a) of Section 1.2, since the level-[math] components partition the vertices of $G$ and every vertex of $G$ is contained in a bag $X_{C}$ corresponding to a level-[math] component $C$. Also, Property (b) is given by the bags $X_{C}$ for level-[math] components $C$, since for every edge $e$ of $G$ one of its endpoints $u$ is a hub of $\operatorname{\textsc{spc}}(r_{i})$ where $i$ is such that $w(e)\in(r_{i},2r_{i}]$, and the other endpoint $w$ is contained in a level-[math] component $C$, for which $X_{C}$ contains $u$ and $w$.

For Property (c), first consider a vertex $u$ of $G$, which is not contained in any set of interface points for any level-$i$ component and any $0\leq i\leq\log_{2}(\alpha)$. Such a vertex only appears in the bag $X_{C}$ for the level-[math] component $C$ containing $u$, and thus the node $v_{C}$ for which the bag contains $u$ trivially induces a connected subtree of $D$.

Any other vertex $u$ of $G$ is an interface point. Let $i$ be the highest level for which $u\in I_{C}$ for some level-$i$ component $C$. We claim that $u\in C$, which implies that $C$ is the unique level-$i$ component containing $u$ in its interface. To show our claim, assume $u\notin C$. Then, by definition, $I_{C}$ contains $u$ because $u\in\operatorname{\textsc{spc}}(r_{j})$ for some $j\geq i$ and $u$ has some neighbour at distance at most $2r_{j}$ in $C$. Since we preprocessed the graph such that every edge is a shortest path between its endpoints, this means that there must be an edge $e=\{u,v\}$ with $w(e)\in(r_{j},2r_{j}]$ and $v\in C$. Since $u\notin C$, we have $i<j$. Let $C^{\prime}$ be the unique level-$j$ component with $C\subseteq C^{\prime}$. Then, by definition, $u\in I_{C^{\prime}}$, which contradicts the maximality of $i$. This proves our claim and shows that the highest level component $C$ with $u\in X_{C}$ is uniquely defined. Moreover, we obtain $u\in\operatorname{\textsc{spc}}(r_{i})$.

Now consider a level-$i^{\prime}$ component $C^{\prime}$ with $i^{\prime}<i$, such that $u\in X_{C^{\prime}}$, and let $C^{\prime\prime}$ be the unique level-$(i^{\prime}+1)$ component containing $C^{\prime}$. We claim that $u\in X_{C^{\prime\prime}}$. If $u\in C^{\prime}\subseteq C^{\prime\prime}$, then $u\in X_{C^{\prime\prime}}$, since $u\in\operatorname{\textsc{spc}}(r_{i})$, $\operatorname{dist}_{C^{\prime\prime}}(u)=0\leq 2r_{i}$ and $i^{\prime}+1\leq i$. If $u\notin C^{\prime}$, then $u\in X_{C^{\prime}}$ implies $u\in I_{C^{\prime}}$, which means that there must be a vertex $w\in C^{\prime}$ with $\operatorname{dist}(u,w)\leq 2r_{i}$. But then $w\in C^{\prime\prime}$ and thus $\operatorname{dist}_{C^{\prime\prime}}(u)\leq 2r_{i}$. Together with $u\in\operatorname{\textsc{spc}}(r_{i})$, this implies $u\in X_{C^{\prime\prime}}$, as claimed. Since $v_{C^{\prime}}$ is a child of $v_{C^{\prime\prime}}$ in the tree $D$, it follows inductively that the nodes of $D$ with bags containing $u$ induce a subtree of $D$ with root $v_{C}$, which establishes Property (c).

By Section 2 each set of interface points contains at most one hub of each level. Since all edges have length at least $3$, there are no hubs in $\operatorname{\textsc{spc}}(r_{0})$ on level [math]. This means that each bag of the tree decomposition contains at most $1+\lceil\log_{2}(\alpha)\rceil$ interface points. The bags for level-[math] components contain one additional vertex. Thus the treewidth of $G$ is at most $1+\lceil\log_{2}(\alpha)\rceil$, as claimed. ∎

An additional property that we will exploit for our algorithms is the following. A $(\mu,\delta)$-net $N\subseteq V$ is a subset of vertices such that (a) the distance between any two distinct net points $u,w\in N$ is more than $\mu$, and (b) for every vertex $v\in V$ there is some net point $w\in N$ at distance at most $\delta$. For graphs of highway dimension $1$ however, we can obtain nets with additional favourable properties, as the next lemma shows.

**Lemma **\thelem

For any graph $G$ of highway dimension $1$ and any $r>0$, there is an $(r,3r)$-net such that every connected component of $G_{\leq r}$ contains exactly one net point. Moreover this net can be computed in polynomial time.

Proof

We first derive an upper bound of $3r$ for the diameter of any connected component of $G_{\leq r}$. Section 2 implies that a connected component $C$ contains at most one hub $x$ of $\operatorname{\textsc{spc}}(r/2)$. By definition, any shortest path in $C$ of length in $(r/2,r]$ must pass through $x$. We also know that every edge of $C$ has length at most $r$. Consequently, every edge in $C$ not incident to $x$ must have length at most $r/2$, since each edge constitutes a shortest path between its endpoints. This implies that any shortest path in $C$ that is not hit by $x$ must have length at most $r/2$: if $C$ contains a shortest path $P$ with length more than $r/2$ not containing $x$ we could repeatedly remove edges of length at most $r/2$ from $P$ until we obtain a shortest path of length in $(r/2,r]$ not hit by $x$, a contradiction. Now consider a shortest path $P$ in $G$ of length more than $r/2$ from some vertex $v\in C$ to $x$ (note that this path may not be entirely contained in $C$). Let $\{u,w\}$ be the unique edge of $P$ such that $\operatorname{dist}(v,u)\leq r/2$ and $\operatorname{dist}(v,w)>r/2$. If the length of the edge $\{u,w\}$ is at most $r/2$ then $\operatorname{dist}(v,w)\leq r$, and thus $w=x$, since the part of the path from $v$ to $w$ is a shortest path of length in $(r/2,r]$ and thus needs to pass through $x$. Otherwise the length of the edge $\{u,w\}$ is in the interval $(r/2,r]$, which again implies $w=x$, since the edge must contain $x$. In either case, $\operatorname{dist}(v,x)\leq 3r/2$. This implies that every vertex in $C$ is at distance at most $3r/2$ from $x$, and thus the diameter of $C$ is at most $3r$.

To compute the $(r,3r)$-net, we greedily pick an arbitrary vertex of each connected component of $G_{\leq r}$. As the distances between components of $G_{\leq r}$ is greater than $r$, and every vertex lies in some component containing a net point, we get the desired distance bounds. Clearly this net can be computed in polynomial time. ∎

3 Approximation schemes

In general the aspect ratio of a graph may be exponential in the input size. A key ingredient of our algorithms is to reduce the aspect ratio $\alpha$ of the input graph $G=(V,E)$ to a polynomial. For STP and TSP, standard techniques can be used to reduce the aspect ratio to $O(n/{\varepsilon})$ when aiming for a $(1+{\varepsilon})$-approximation. This was for instance also used in [Fel+18] for low highway dimension graphs, but here we need to take special care not to destroy the structural properties given by Section 2 in this process. In particular, we need to reduce the aspect ratio and maintain the fact that the treewidth is bounded.

Therefore, we reduce the aspect ratio of our graphs by the following preprocessing. Both metric TSP and STP admit constant factor approximations in polynomial time using well-known algorithms [Byr+10, Chr76]. We first compute a solution of cost $c$ using a $\beta$-approximation algorithm for the problem at hand (TSP or STP). For TSP, the diameter of the graph $G$ clearly is at most $c/2$. For STP we remove every vertex of $V$ that is at distance more than $c$ from any terminal, since such a vertex cannot be part of the optimum solution. After having removed all such vertices in this way, we obtain a graph $G$ of diameter at most $3c$. Thus, in the following, we may assume that our graph $G$ has diameter at most $3c$. We then set $r=\frac{{\varepsilon}c}{3n}$ in Section 2 to obtain a $(\frac{{\varepsilon}c}{3n},\frac{{\varepsilon}c}{n})$-net $N\subseteq V$. As a consequence the metric induced by $N$ (with distances of $G$) has aspect ratio at most $\frac{3c}{{\varepsilon}c/(3n)}=O(n/{\varepsilon})$, since the minimum distance between any two net points of $N$ is at least $\frac{{\varepsilon}c}{3n}$ and the maximum distance is at most $3c$. We will exploit this property in the following.

By Section 2, each connected component of $G_{\leq\frac{{\varepsilon}c}{3n}}$ contains exactly one net point of $N$. Let $\eta\colon V\mapsto N$ map each vertex of $G$ to the unique net point in the same connected component of $G_{\leq\frac{{\varepsilon}c}{3n}}$. We define a new graph $G^{\prime}$ with vertex set $N\subseteq V$ and edge set $\{\{\eta(u),\eta(v)\}:\{u,v\}\in E\land\eta(u)\neq\eta(v)\}$. The length of each edge $\{w,w^{\prime}\}$ of $G^{\prime}$ is the shortest path distance between $w$ and $w^{\prime}$ in $G$. This new graph $G^{\prime}$ may not have bounded highway dimension, but we claim that it has treewidth $O(\log(n/{\varepsilon}))$.

**Lemma **\thelem

If $G$ has highway dimension $1$, the graph $G^{\prime}$ with vertex set $N$ has treewidth $O(\log(n/{\varepsilon}))$. Moreover, a tree decomposition for $G^{\prime}$ of width $O(\log(n/{\varepsilon}))$ can be computed in polynomial time.

Proof

We construct a tree decomposition $D^{\prime}$ of $G^{\prime}$ as follows. Following Section 2 we can compute a tree decomposition $D$ of width at most $1+\lceil\log_{2}(\alpha)\rceil$, where $\alpha$ is the aspect ratio of $G$: for this we need to compute a locally $1$-sparse shortest path cover $\operatorname{\textsc{spc}}(r_{i})$ for each level $i$, which can be done in polynomial time via an XP algorithm [Fel+18] if the highway dimension is $1$. We then find the level-$i$ components and their interface points, from which the tree decomposition $D$ and its bags can be constructed. Since there are $O(\log\alpha)$ levels and $\alpha$ is at most exponential in the input size (which includes the encoding length of the edge weights), we can compute $D$ in polynomial time.

We construct $D^{\prime}$ from $D$ by replacing every bag $X$ of $D$ by a new bag $X^{\prime}=\{\eta(v):v\in X\}$ containing the net points for the vertices in $X$. It is not hard to see that Properties (a) and (b) of Section 1.2 are fulfilled by $D^{\prime}$, since they are true for $D$. For Property (c), note that for any edge $\{u,v\}$ of $G$, the set of all bags of $D$ that contain $u$ or $v$ form a connected subtree of $D$. This is because the bags containing $u$ form a connected subtree (Property (c)), the same is true for $v$, and both these subtrees share at least one node labelled by a bag containing the edge $\{u,v\}$ (Property (b)). Consequently, the set of all bags containing vertices of any connected subgraph of $G$ form a connected subtree. In particular, for any connected component $A$ of $G_{\leq\frac{{\varepsilon}c}{3n}}$, the set of bags of $D$ containing at least one vertex of $A$ form a connected subtree. This implies Property (c) for $D^{\prime}$. Thus, $D^{\prime}$ is indeed a tree decomposition of $G^{\prime}$ according to Section 1.2. Note that $D^{\prime}$ can be computed in polynomial time.

To bound the width of $D^{\prime}$, recall that a bag $X$ of the tree decomposition $D$ of $G$ contains the interface points $I_{C}$ of a level-$i$ component $C$, in addition to one more vertex of $C$ on the lowest level $i=0$. Each interface point is a hub from $\operatorname{\textsc{spc}}(r_{j})$ at some level $j\geq i$ and is at distance at most $2r_{j}$ from $C$. In particular, if $2r_{i}\leq\frac{{\varepsilon}c}{3n}$ then $C$ is a component of $G_{\leq 2r_{i}}\subseteq G_{\leq\frac{{\varepsilon}c}{3n}}$, and all hubs of $I_{C}\cap\operatorname{\textsc{spc}}(r_{j})$ for which $2r_{j}\leq\frac{{\varepsilon}c}{3n}$ lie in the same connected component $A$ of $G_{\leq\frac{{\varepsilon}c}{3n}}$ as $C$. These hubs are therefore all mapped to the same net point $w$ in $A$ by $\eta$. In addition to $w$, the bag $X^{\prime}=\{\eta(v):v\in X\}$ resulting from $X$ and $\eta$ contains at most one vertex for every level $j$ such that $2r_{j}>\frac{{\varepsilon}c}{3n}$. As $r_{j}=2^{j}$, this condition is equivalent to $j>\log_{2}(\frac{{\varepsilon}c}{3n})-1$. As there are $1+\lceil\log_{2}(\alpha)\rceil$ levels in total, there are $O(\log(\frac{\alpha n}{{\varepsilon}c}))$ hubs in $X^{\prime}$. This bound is obviously also valid in case $2r_{i}>\frac{{\varepsilon}c}{3n}$. We preprocessed the graph $G$ so that its diameter is at most $3c$ and its minimum distance is $3$, which implies an aspect ratio $\alpha$ of at most $c$ for $G$. This means that every bag $X^{\prime}$ contains $O(\log(n/{\varepsilon}))$ vertices, and thus the claimed treewidth bound for $G^{\prime}$ follows. ∎

We are now ready to prove our main result.

Proof (of Theorem 1)

To solve TSP or STP on $G$ we first use the above reduction to obtain $G^{\prime}$ and its tree decomposition $D^{\prime}$, and then compute an optimum solution for $G^{\prime}$. For TSP, $G^{\prime}$ is already a valid input instance, but for STP we need to define a terminal set, which simply is $R^{\prime}=\{\eta(v)\mid v\in R\}$ if $R$ is the terminal set of $G$. [Bod+13] proved that for both TSP and STP there are deterministic algorithms to solve these problems exactly in time $2^{O(t)}n$, given a tree decomposition of the input graph of width $t$. By Section 3 we can thus compute the optimum to $G^{\prime}$ in time $2^{O(\log(n/{\varepsilon}))}\cdot n=(n/{\varepsilon})^{O(1)}$. Afterwards, we convert the solution for $G^{\prime}$ back to a solution for $G$, as follows.

For TSP we may greedily add vertices of $V$ to the tour on $N$ by connecting every vertex $v\in V$ to the net point $\eta(v)$. As the vertices $N$ of $G^{\prime}$ form a $(\frac{{\varepsilon}c}{3n},\frac{{\varepsilon}c}{n})$-net of $V$, this incurs an additional cost of at most $2\frac{{\varepsilon}c}{n}$ per vertex, which sums up to at most $2{\varepsilon}c$. Let Opt and $\textsc{Opt}^{\prime}$ denote the costs of the optimum tours in $G$ and $G^{\prime}$, respectively. We know that $c\leq\beta\cdot\textsc{Opt}$, since we used a $\beta$-approximation algorithm to compute $c$. Furthermore, the optimum tour in $G$ can be converted to a tour in $G^{\prime}$ of cost at most Opt by short-cutting, due to the triangle inequality. Thus $\textsc{Opt}^{\prime}\leq\textsc{Opt}$, which means that the cost of the computed tour in $G$ is at most $\textsc{Opt}^{\prime}+2{\varepsilon}c\leq(1+2\beta{\varepsilon})\textsc{Opt}$.

Similarly, for STP we may greedily connect a terminal $v$ of $G$ to the terminal $\eta(v)$ of $G^{\prime}$ in the computed Steiner tree in $G^{\prime}$. This adds an additional cost of at most $\frac{{\varepsilon}c}{n}$, which sums up to at most ${\varepsilon}c$. Let now Opt and $\textsc{Opt}^{\prime}$ be the costs of the optimum Steiner trees in $G$ and $G^{\prime}$, respectively. We may convert a Steiner tree $T$ in $G$ into a tree $T^{\prime}$ in $G^{\prime}$ by using edge $\{\eta(u),\eta(v)\}$ for each edge $\{u,v\}$ of $T$. Note that the resulting tree $T^{\prime}$ contains all terminals of $G^{\prime}$, since $R^{\prime}=\{\eta(v)\mid v\in R\}$. As the vertices $N$ of $G^{\prime}$ form a $(\frac{{\varepsilon}c}{3n},\frac{{\varepsilon}c}{n})$-net of $V$, the cost of $T^{\prime}$ is at most $\textsc{Opt}+2{\varepsilon}c$ if the cost of $T$ is Opt (by the same argument as used for the proof of Section 3). As before, we know that $c\leq\beta\cdot\textsc{Opt}$, and thus the cost of the computed Steiner tree in $G$ is at most $\textsc{Opt}^{\prime}+{\varepsilon}c\leq\textsc{Opt}+3{\varepsilon}c\leq(1+3\beta{\varepsilon})\textsc{Opt}$.

Hence we obtain FPTASs for both TSP and STP, which compute $(1+{\varepsilon})$-approximations within a runtime that is polynomial in the input size and $1/{\varepsilon}$. ∎

We prove next that STP is $\mathsf{NP}$-hard on graphs of highway dimension $1$, which means that the problem is weakly $\mathsf{NP}$-hard for these inputs (cf. [Vaz01]). Whether TSP is $\mathsf{NP}$-hard for such small highway dimension remains open, but we prove that it is for highway dimension $6$.

4 Hardness of Steiner Tree for highway dimension 1

We present a reduction from the $\mathsf{NP}$-hard satisfiability problem (SAT) [GJ02], in which a Boolean formula $\varphi$ in conjunctive normal form is given, and a satisfying assignment of its variables needs to be found.

Proof (of Theorem 2)

For a given SAT formula $\varphi$ with $k$ variables and $\ell$ clauses we construct a graph $G_{\varphi}$ as follows (cf. Fig. 1). For each variable $x$ we introduce a path $P_{x}=(t_{x},u_{x},f_{x})$ with two edges of length $1$ each. The vertex $u_{x}$ is a terminal. Additionally we introduce a terminal $v_{0}$, which we call the root, and add the edges $\{v_{0},t_{x}\}$ and $\{v_{0},f_{x}\}$ for every variable $x$. Every edge incident to $v_{0}$ has length $11$. For each clause $C_{i}$, where $i\in\{1,\ldots,\ell\}$, we introduce a terminal $v_{i}$ and add the edge $\{v_{i},t_{x}\}$ for each variable $x$ such that $C_{i}$ contains $x$ as a positive literal, and we add the edge $\{v_{i},f_{x}\}$ for each $x$ for which $C_{i}$ contains $x$ as a negative literal. Every edge incident to $v_{i}$ has length $11^{i+1}$. Note that the edges incident to the root $v_{0}$ also have length $11^{i+1}$ for $i=0$.

**Lemma **\thelem

The constructed graph $G_{\varphi}$ has highway dimension $1$.

Proof

Fix a scale $r>0$. If $r\leq 5$ then the shortest path cover $\operatorname{\textsc{spc}}(r)$ only needs to hit shortest paths of length at most $2r\leq 10$. Since all edges incident to terminals $v_{j}$ with $j\in\{0,\dots,\ell\}$ have length at least $11$, any such path contains only edges of paths $P_{x}$. Thus it suffices to include all vertices $u_{x}$ in $\operatorname{\textsc{spc}}(r)$. A ball $B_{w}(2r)$ of radius $2r\leq 10$ can also only contain some subset of vertices of a single path $P_{x}$, or a single vertex $v_{j}$. In the former case the ball contains at most the vertex $u_{x}\in\operatorname{\textsc{spc}}(r)$, and in the latter none of $\operatorname{\textsc{spc}}(r)$.

If $r>5$, let $i=\lfloor\log_{11}(r/5)\rfloor\geq 0$ and $\operatorname{\textsc{spc}}(r)=\{v_{i}\}$. Since there is only one hub, this shortest path cover is locally $1$-sparse. Note that any edge incident to a vertex $v_{j}$ with $j\geq i+1$ has length at least $11^{i+2}\geq 11r/5>2r$. Also, all paths that do not use any $v_{j}$ with $j\geq i$ have length at most $2+\sum_{j=0}^{i-1}(2\cdot 11^{j+1}+2)$, since such a path can contain at most two edges incident to a vertex $v_{j}$ with $j\leq i-1$ and the paths $P_{x}$ of length $2$ are connected only through edges incident to vertices $v_{j}$. The length of such a path is thus shorter than {doitall} 2+211i+111-1+2i≤3⋅11^i+2⋅11^i ≤5⋅11^i ≤r,

where the first inequality holds since $i+1\leq 11^{i}$ whenever $i\geq 0$. Hence the only paths that need to be hit by hubs on scale $r$ are those passing through $v_{i}$, which is a hub of $\operatorname{\textsc{spc}}(r)$. ∎

To finish the reduction, we claim that there is a satisfying assignment for $\varphi$ if and only if there is a Steiner tree $T$ for $G_{\varphi}$ with cost at most $12k+\sum_{i=1}^{\ell}11^{i+1}$. If there is a satisfying assignment for $\varphi$, then the tree $T$ contains the edges $\{u_{x},t_{x}\}$ and $\{v_{0},t_{x}\}$ for variables $x$ that are set to true, and the edges $\{u_{x},f_{x}\}$ and $\{v_{0},f_{x}\}$ for variables $x$ that are set to false. This connects every terminal $u_{x}$ with the root $v_{0}$, and the cost of these edges is $12k$. For every terminal $v_{i}$ where $i\geq 1$ we can now add the edge $\{v_{i},s_{x}\}$ for $s_{x}\in\{t_{x},f_{x}\}$ that corresponds to a literal of $C_{i}$ that is true in the satisfying assignment. Since this Steiner vertex $s_{x}$ is connected to the root $v_{0}$, we obtain a Steiner tree $T$. The latter edges add another $\sum_{i=1}^{\ell}11^{i+1}$ to the solution cost, and thus the total cost is as claimed.

Conversely, consider a minimum cost Steiner tree $T$ in $G_{\varphi}$. Note that for any terminal $u_{x}$ the tree must contain an incident edge of cost $1$, while for any terminal $v_{i}$ with $i\geq 1$ the tree must contain an incident edge of cost $11^{i+1}$. This adds up to a cost of $k+\sum_{i=1}^{\ell}11^{i+1}$. Assume that there is some variable $x$ such that $T$ contains neither $\{v_{0},t_{x}\}$ nor $\{v_{0},f_{x}\}$. This means that in $T$ the terminal $u_{x}$ is connected to the root $v_{0}$ through an edge $\{v_{i},s_{x}\}$ for $s_{x}\in\{t_{x},f_{x}\}$ and some $i\geq 1$. The edge $v_{0}s_{x}$ forms a fundamental cycle with the tree $T$, which however has a shorter length of $11$ compared to the edge $\{v_{i},s_{x}\}$, which has length $11^{i+1}$. Thus removing $\{v_{0},s_{x}\}$ and adding $\{v_{i},s_{x}\}$ instead, would yield a cheaper Steiner tree. As this would contradict that $T$ has minimum cost, $T$ contains at least one of the edges $\{v_{0},t_{x}\}$ and $\{v_{0},f_{x}\}$ for every variable $x$. This adds another $11k$ to the cost, so that $T$ costs at least $12k+\sum_{i=1}^{\ell}11^{i+1}$.

If we assume that $12k+\sum_{i=1}^{\ell}11^{i+1}$ is also an upper bound on the cost of $T$, by the above observations the tree $T$ contains exactly one edge incident to every terminal $u_{x}$ and $v_{i}$ for $i\geq 1$, and exactly $k$ edges incident to $v_{0}$. Furthermore, for every variable $x$ the latter edges contain exactly one of $\{v_{0},t_{x}\}$ and $\{v_{0},f_{x}\}$. Thus $T$ encodes a satisfying assignment for $\varphi$, as follows. For every edge $v_{0}t_{x}$ we may set $x$ to true, and for every edge $\{v_{0},f_{x}\}$ we may set $x$ to false. For every clause $C_{i}$ the corresponding terminal $v_{i}$ connects through one of the Steiner vertices $s_{x}\in\{t_{x},f_{x}\}$ of a corresponding literal contained in $C_{i}$. The only incident vertices to $s_{x}$ in $G_{\varphi}$ are some terminals $v_{j}$, the terminal $u_{x}$, and the root $v_{0}$. As each $v_{j}$ and also $u_{x}$ only has one incident edge contained in the tree $T$, the tree must contain the edge $\{v_{0},s_{x}\}$ so that the root can be reached from $s_{x}$ in $T$. Hence $s_{x}$ corresponds to a literal that is true in $C_{i}$. Using Section 4, which bounds the highway dimension of $G_{\varphi}$, we obtain Theorem 2. ∎

5 Hardness of Travelling Salesperson for highway dimension 6

We now show hardness of TSP for graphs of bounded highway dimension. We first introduce a simple lemma that will allow us to easily bound the highway dimension of our construction by adding edges incrementally. In the following we denote the highway dimension of a graph $G$ by $\operatorname{hd}(G)$.

**Definition **\thedfn

A cost $c^{\star}\in\mathbb{R}$ is safe w.r.t. a (multi-)set of costs $C\subseteq\mathbb{R}$ if $c^{\star}\geq 2\sum_{c\in C}c$.

**Lemma **\thelem

Let $G=(V,E)$ be a graph, $E^{\prime}\subseteq{V\choose 2}$, and $G^{\prime}=(V,E\cup E^{\prime})$. If the edges in $E^{\prime}$ have safe costs w.r.t. the edge costs of $G$, then $\operatorname{hd}(G^{\prime})\leq\max\{\operatorname{hd}(G),|E^{\prime}|\}$.

Proof

Let $c^{\star}$ be the smallest cost among the edges of $E^{\prime}$ and consider a fixed scale $r\in\mathbb{R}^{+}$. If $r<c^{\star}/2$, then no path of length $l\in(r,2r]$ in $G^{\prime}$ contains any of the edges in $E^{\prime}$. By the definition of the highway dimension of $G$, there is a locally $\operatorname{hd}(G)$-sparse shortest path cover $\textsc{spc}(r)$ of $G^{\prime}$ for this scale. Now if $r\geq c^{\star}/2$, then every path of length $l\in(r,2r]$ in $G^{\prime}$ must contain an edge of $E^{\prime}$, since the costs of all edges in $E$ sum up to at most $r$. Therefore, we can find a shortest path cover $\textsc{spc}(r)$ of $G^{\prime}$ simply by taking a minimum vertex cover of $E^{\prime}$, which has size at most $|E^{\prime}|$. ∎

We are now ready to prove hardness.

Proof (of Theorem 3.)

We reduce from $(\leq\!3,3)\textrm{-}\textsc{Sat}$ [GJ02]. To that end, let a $(\leq\!3,3)\textrm{-}\textsc{Sat}$ formula be given, with variables $x_{1},\dots x_{n}$ and clauses $C_{1},\dots,C_{m}$, where each literal appears at most twice (and each variable at most three times). We construct a graph $G$ with edge costs taken from among the values $a\ll b\ll c_{1}\ll\dots\ll c_{n-1}\ll d\ll e\ll f_{1}\ll\dots\ll f_{m}$, where each cost value can be chosen arbitrarily such that it is safe with respect to the costs of all cheaper edges. For example, there will be $2n$ edges of cost $a$, hence we choose $b\geq 4an$. Let $T^{\star}$ be (any) TSP tour in $G$ of minimum cost $|T^{\star}|$. We consider $T^{\star}$ to be oriented arbitrarily in one of its two possible orientations.

For every variable $x_{i}$, we introduce a gadget with four vertices $v_{i1},v_{i2},v_{i3},v_{i4}$ and the edges $\{v_{i1},v_{i3}\}$ and $\{v_{i2},v_{i4}\}$, both of cost $a$. We further add edges $\{v_{i1},v_{i2}\},\{v_{i2},v_{i3}\},\{v_{i3},v_{i4}\},\{v_{i4},v_{i1}\}$ of cost $b$ each (cf. Fig. 2). We will enforce that $T^{\star}$ uses both edges of cost $a$ in every variable gadget, and we will interpret the variable as ’true’, if the orientation of these edges along the tour is $(v_{i1},v_{i3})$ and $(v_{i4},v_{i2})$, or $(v_{i3},v_{i1})$ and $(v_{i2},v_{i4})$, and ’false’ otherwise. Let $G^{b}$ be the graph we have constructed so far. Each of the $n$ components of $G^{b}$ has highway dimension $2$: For scales $r<a$, we can set $\textsc{spc}(r)=\bigcup_{i}\{v_{i1},v_{i2}\}$, and for scales $r\geq a$, we can set $\textsc{spc}(r)=\bigcup_{i}\{v_{i1},v_{i3}\}$. Hence, $\operatorname{hd}(G^{b})=2$.

We connect the variable gadgets by adding edges $\{v_{i1},v_{(i+1)1}\}$ of cost $c_{i}$ for all $i\in\{1,\dots,n-1\}$ in this order. By definition, each new edge has a safe cost (w.r.t. all previous edges), and hence, by Section 5, the resulting graph $G^{c}$ has highway dimension $\operatorname{hd}(G^{2})=2$. We will enforce that $T^{\star}$ uses each of these edges exactly twice.

For each clause $C_{j}$, we introduce a clause gadget with six vertices $w_{j1}$, $w^{\prime}_{j1}$, $w_{j2}$, $w_{j2}^{\prime}$, $w_{j3}$, $w_{j3}^{\prime}$ (cf. Fig. 3). We first add three edges $\{w_{j1}^{\prime},w_{j2}\}$, $\{w_{j2}^{\prime},w_{j3}\}$, $\{w_{j3}^{\prime},w_{j1}\}$ of cost $d$ each. Since these edges are disconnected, the resulting graph $G^{d}$ still has $\operatorname{hd}(G^{d})=2$. Now, we add three edges $\{w_{j1},w_{j1}^{\prime}\}$, $\{w_{j2},w_{j2}^{\prime}\}$, $\{w_{j3},w_{j3}^{\prime}\}$ of cost $e$ each. In the resulting graph $G^{e}$, clauses are still disconnected. Since we added three edges with safe costs for each clause, by Section 5, we have $\operatorname{hd}(G^{e})\leq 3$.

Finally, we connect each clause gadget for clause $C_{j}$ to the three variable gadgets corresponding to the variables appearing in $C_{j}$ (cf. Fig. 3). To this end, we add six edges per clause, step by step in the order of increasing clause indices. Let $C_{j}=\lambda_{j1}\vee\lambda_{j2}\vee\lambda_{j3}$ and consider $k\in\{1,2,3\}$. Assume $\lambda_{jk}=x_{i}$, i.e., $x_{i}$ appears as a positive literal in $C_{j}$. Let $\delta=0$ if $C_{j}$ is the first clause containing the literal $x_{i}$, i.e., $x_{i}\notin C_{j^{\prime}}$ for $j^{\prime}<j$, and $\delta=2$ otherwise. We add the edges $\{w_{jk},v_{i(2+\delta)}\},\{w_{jk}^{\prime},v_{i(1+\delta)}\}$ of cost $f_{j}$. Now assume $\lambda_{jk}=\bar{x}_{i}$ and let again $\delta=0$ if $\bar{x}_{i}\notin C_{j^{\prime}}$ for $j^{\prime}<j$, and $\delta=2$ otherwise. We add the edges $\{w_{jk},v_{i(3-\delta)}\},\{w_{jk}^{\prime},v_{i(2+\delta)}\}$ of cost $f_{j}$. Since we add six edges of safe costs in each step, by Section 5, the final graph $G=G^{f}$ has $\operatorname{hd}(G)\leq 6$.

Now let $W=2\sum_{j=1}^{m}f_{j}+2me+3md+2\sum_{i=1}^{n-1}c_{i}+(2n-m)b+2na$. We claim that $|T^{\star}|\leq W$ if and only if the $(\leq\!3,3)\textrm{-}\textsc{Sat}$ formula is satisfiable.

For the first part of the claim, assume the $(\leq\!3,3)\textrm{-}\textsc{Sat}$ formula is satisfiable, and, for all $j\in\{1,\dots,m\}$, let $y_{j}$ be a unique variable that satisfies clause $C_{j}$ in the corresponding assignment. We describe a tour of cost $W$ by constructing a Eulerian graph consisting of edges of $G$ (sometimes twice) that connect all vertices with a total cost of $W$. We start by including the cycle of cost $3e+3d$ within each clause gadget, and each edge between different variable gadgets twice, for a total cost of $3me+3md+2\sum_{i}^{n-1}c_{i}$. For every variable $x_{i}$ that is set to true, we include the cycle $v_{i1},v_{i3},v_{i4},v_{i2},v_{i1}$ of cost $2b+2a$. For every variable $x_{i}$ that is set to false, we include the cycle $v_{i1},v_{i3},v_{i2},v_{i4},v_{i1}$ of cost $2b+2a$. The resulting graph $T^{\prime}$ is Eulerian, since we added only cycles, but not yet connected. Its cost is $3me+3md+2\sum_{i=1}^{n-1}c_{i}+2nb+2na$.

Now take a clause $C_{j}$ that is satisfied by variable $x_{i}=y_{j}$. We add the edges $\{w_{jk},v_{ir}\},\{w_{jk}^{\prime},v_{i(r+3\mod 4)}\}$ that connect the corresponding clause and variable gadgets. Observe that the edge $\{w_{jk},w_{jk}^{\prime}\}$ is in $T^{\prime}$ and so is the edge $\{v_{ir},v_{i(r+3\mod 4)}\}$, because $x_{i}$ satisfies $C_{j}$. We can thus remove these two edges and obtain a Eulerian graph. This increases the cost of the graph by $2f_{j}-e-b$. The final graph $T$ is connected, Eulerian, and has cost $W$ as claimed.

For the second part of the claim, consider any TSP tour $T$ with $|T|\leq W$. Observe that $3f_{m}>W$, $3f_{m-1}>W-2f_{m}$, and so on. Since, for all $j\in\{1,\dots,m$}, the edges of cost $f_{j}$ form a cut of $G$, we can conclude that $T$ uses exactly 2 edges of cost $f_{j}$ (or one of them twice). Similarly, $(2m+1)e>W-\sum_{j=1}^{n-1}f_{j}$, but $T$ needs to use at least two edges of cost $e$ (or one of them twice) to connect all vertices of a clause gadget for $C_{j}$ to the two edges of cost $f_{j}$ that are part of the tour. We can again conclude that $T$ uses exactly two edges of cost $e$ in each clause gadget. And again $(3m+1)d>W-\sum_{j=1}^{n-1}f_{j}-2me$, but $T$ needs to use at least three times an edge of cost $d$ to connect all vertices of a clause gadget, provided that it uses only two edges of cost $e$. We conclude that $T$ uses exactly three edges of cost $d$ in each clause gadget. Finally, observe that the only way to connect all vertices of a clause gadget with two edges of cost $e$ and three edges of cost $d$ needs that the two edges of cost $f_{j}$ are distinct and connect to the same vertex gadget. We will rely on this observation in the following, and on the fact that costs are chosen to be safe with respect to all smaller costs (in particular, $e$ is safe with respect to $a,b,c,d$, etc.).

The first implication is that $T$ needs to use every edge between vertex gadgets twice, since the vertex gadgets are not connected via clause gadgets. Our analysis so far implies that edges within variable gadgets that are used in $T$ incur a cost of at most $W^{\prime}=(2n-m)b+2na$. Let $k\in\{0,\dots 4\}$ be the number of clause gadgets in $T$ connected to the variable gadget of $x_{i}$. Clearly, $T$ needs to use at least $4-k$ edges within the variable gadget. The cost within a variable gadget depending on $k$ is at least $2b+2a$ (if $k=0$), $b+2a$ (if $k=1$), $2a$ (if $k=2$), $a$ (if $k=1$), or 0 (if $k=4$). Since each variable appears in at most four clauses and each clause has at most 3 literals, we have $m\leq 4n/3<2n$. To obtain a cost of at most $W^{\prime}$, we must thus have $k\leq 2$ in each variable gadget, since $b\gg a$. Furthermore, if there are two clause gadgets connected to a variable gadget, they must connect to disjoint vertices of the clause to allow for a cost of at most $2a$ in the variable gadget. This means that the corresponding literals must either both be positive or both be negative. But if there is an assignment of clauses to variables such that at most two clauses are assigned to each variable and the corresponding literal of the assigned clauses must agree, this immediately yields a satisfying assignment of the $(\leq\!3,3)\textrm{-}\textsc{Sat}$-formula. ∎

6 Conclusions

We showed that, somewhat surprisingly, graphs of highway dimension $1$ exhibit a rich combinatorial structure. On one hand, it was already known [Fel+18] that these graphs are not minor-closed and thus their treewidth is unbounded. Here we additionally showed that STP is weakly $\mathsf{NP}$-hard on such graphs, further confirming that these graphs have non-trivial properties. On the other hand, we proved in Section 2 that the treewidth of a graph of highway dimension $1$ is logarithmically bounded in the aspect ratio $\alpha$. This in turn can be exploited to obtain a very efficient FPTAS for both STP and TSP.

At this point one may wonder whether it is possible to generalize Section 2 to larger values of the highway dimension. In particular, in [Fel+18] it was suggested that the treewidth of a graph of highway dimension $h$ might be bounded by, say, $O(h\,\textrm{polylog}(\alpha))$. However such a bound is highly unlikely in general, since it would have the following consequence for the $k$-Center problem, for which $k$ vertices (centers) need to be selected in a graph such that the maximum distance of any vertex to its closest center is minimized. It was shown in [Fel18] that it is $\mathsf{NP}$-hard to compute a $(2-{\varepsilon})$-approximation for $k$-Center on graphs of highway dimension $O(\log^{2}n)$, for any ${\varepsilon}>0$. Given such a graph, the same preprocessing of Section 3 could be used to derive an analogue of Section 3, i.e., a graph $G^{\prime}$ of treewidth $O(\textrm{polylog}(n/{\varepsilon}))$ could be computed for the net $N$. Moreover, a $2$-approximation for $k$-Center can be computed in polynomial time on any graph [HS86], and if the input has treewidth $t$ a $(1+{\varepsilon})$-approximation can be computed in $(t/{\varepsilon})^{O(t)}n^{O(1)}$ time [KLP17]. Using the same arguments to prove Theorem 1 for STP and TSP, it would now be possible to compute a $(1+{\varepsilon})$-approximation for $k$-Center in quasi-polynomial time (cf. [FM18]). That is, we would obtain a QPTAS for graphs of highway dimension $O(\log^{2}n)$, which is highly unlikely given that computing a $(2-{\varepsilon})$-approximation is $\mathsf{NP}$-hard on such graphs.

The above argument rules out any bound of $(h\log\alpha)^{O(1)}$ for graphs of highway dimension $h$ and aspect ratio $\alpha$, unless $\mathsf{NP}$-hard problems admit quasi-polynomial time algorithms. In fact, we conjecture that the $k$-Center problem is $\mathsf{NP}$-hard to approximate within a factor of $2-{\varepsilon}$ for graphs of constant highway dimension (for some constant larger than $1$). If this is true, then the above argument even rules out a treewidth bound of $f(h)\,\textrm{polylog}(\alpha)$ for any function $f$. Thus, in order to answer the open problem of [Fel+18] and obtain a PTAS for graphs of constant highway dimension, a different approach seems to be needed.