Hierarchy of Transportation Network Parameters and Hardness Results
HHierarchy of Transportation Network Parametersand Hardness Results
Johannes Blum
University of Konstanz, [email protected]
Abstract
The graph parameters highway dimension and skeleton dimension were introduced to capture theproperties of transportation networks. As many important optimization problems like
TravellingSalesperson , Steiner Tree or k -Center arise in such networks, it is worthwhile to study themon graphs of bounded highway or skeleton dimension.We investigate the relationships between mentioned parameters and how they are related to otherimportant graph parameters that have been applied successfully to various optimization problems.We show that the skeleton dimension is incomparable to any of the parameters distance to linearforest, bandwidth, treewidth and highway dimension and hence, it is worthwhile to study mentionedproblems also on graphs of bounded skeleton dimension. Moreover, we prove that the skeletondimension is upper bounded by the max leaf number and that for any graph on at least three verticesthere are edge weights such that both parameters are equal.Then we show that computing the highway dimension according to most recent definition isNP-hard, which answers an open question stated by Feldmann et al. [18]. Finally we prove that ongraphs G = ( V, E ) of skeleton dimension O (log | V | ) it is NP-hard to approximate the k -Center problem within a factor less than 2. Mathematics of computing → Graph theory; Theory of computa-tion → Problems, reductions and completeness; Theory of computation → Parameterized complexityand exact algorithms
Keywords and phrases
Graph Parameters, Skeleton Dimension, Highway Dimension, k -Center Many important optimization problems arise in the context of road or flight networks, e.g.
Travelling Salesperson or Steiner Tree , and have applications in domains like routeplanning or logistics. Therefore, several approaches have been developed that try to exploitthe special structure of such transportation networks. Examples are the graph parametershighway dimension and skeleton dimension. Intuitively, a graph has low highway dimension hd or skeleton dimension κ , if there is only a limited number of options to leave a certainregion of the network on a shortest path. Both parameters were originally used in theanalysis of shortest path algorithms and it was shown that if hd or κ are small, thereare preprocessing-based techniques to compute shortest paths significantly faster than thealgorithm of Dijkstra [3, 2, 1, 25].The highway dimension was also investigated in the context of NP-hard optimizationproblems, such as Travelling Salesperson ( TSP ), Steiner Tree and
Facility Lo-cation [18], k -Center [17, 20, 10] or k -Median and Bounded-Capacity VehicleRouting [10]. It was shown that in many cases, graphs of low highway dimensions allowbetter algorithms than general graphs. To our knowledge, the skeleton dimension has exclu-sively been studied in the context of shortest path algorithms so far. However, it was shownthat real-world road networks exhibit a skeleton dimension that is clearly smaller than thehighway dimension [11]. Moreover, in contrast to the highway dimension, it can be computedin polynomial time. Hence it is natural to study the aforementioned problems on networksof low skeleton dimension. a r X i v : . [ c s . D M ] N ov Hierarchy of Transportation Network Parameters and Hardness Results
Further graph classes that have been used to model transportation networks are forinstance planar graphs and graphs of low treedwidth or doubling dimension. Moreover, manyimportant optimization problems have been studied extensively for classic graph parameterslike treewidth or pathwidth [12, 4]. Still, there are only partial results on how the highwaydimension hd and skeleton dimension κ are related to these parameters. This is the startingpoint of the present paper. A better understanding of the relationships between hd , κ and different well-studied graph parameters will allow a deeper insight in the structureof transportation networks and might enable further algorithms custom-tailored for suchnetworks. We now briefly sum up some algorithmic results in the context of optimization problems intransportation networks. Arora [5] developed a general framework that enables PTASs forseveral geometric problems where the network is embedded in the Euclidean plane. Buildingupon the work of Arora, Talwar [28] developed QPTASs for
TSP , Steiner Tree , k -Median and Facility Location on graphs of low doubling dimension (for a formal definition, seeDefinition 2). This was improved by Bartal et al. [6], who obtained a PTAS for
TSP . Asthe skeleton dimension of a graph upper bounds its doubling dimension (cf. Section 2.1)the aforementioned results immediately imply a PTAS for
TSP and QPTASs for
SteinerTree , k -Median and Facility Location .The k -Center problem is NP-complete on general graphs [29] and has been subject to ex-tensive research. In fact, for any (cid:15) >
0, it is NP-hard to compute a (2 − (cid:15) )-approximation, evenwhen considering only planar graphs [26], geometric graphs using L or L ∞ distances or graphsof highway dimension O (log | V | )[17]. However, there is a fairly simple 2-approximationalgorithm for general graphs by Hochbaum and Shmoys [23].One way to approximate k -Center better than by a factor of 2 is the use of so called fixed-parameter approximation algorithms (FPAs) . The basic idea is to combine the conceptsof fixed-parameter algorithms and approximation algorithms. Formally, for α >
1, an α -FPAfor a parameter p is an algorithm that computes an α -approximation in time f ( p ) · n O (1) where f is a computable function. Feldmann [17] showed there is a / -FPA for k -Center when parameterizing both by the number of center nodes k and the highway dimension hd .Later, Becker et al. [10] showed that for any (cid:15) > (cid:15) )-FPA for k -Center whenparameterizing by k and hd , using a slightly different definition for the highway dimension asin [17] (see also Section 2.2). Moreover, on graphs of doubling dimension d , it is possible tocompute a (1 + (cid:15) )-approximation in time (cid:0) k k /(cid:15) O ( k · d ) (cid:1) · n O (1) [20]. As the doubling dimensionis a lower bound for the skeleton dimension κ , this implies a (1 + (cid:15) )-FPA for parameter( (cid:15), k, κ ). However, computing a (2 − (cid:15) )-approximation is W [2]-hard when parameterizing onlyby k , and unless the exponential time hypothesis (ETH) fails, it is not possible to compute a(2 − (cid:15) )-approximation in time 2 o ( √ hd ) · n O (1) for highway dimension hd [17]. We first give an overview of various graph parameters, in particular we review several slightlydifferent definitions of the highway dimension that can be found in the literature. Thenwe show relationships between skeleton dimension, highway dimension and other importantparameters. Our results include the following.The max leaf number ml is a tight upper bound for the bandwidth bw . This improves aresult of Sorge et al. who showed that bw ≤ ml [27]. . Blum 3 The skeleton dimension is incomparable to any of the parameters distance to linear forest,bandwidth, treewidth and highway dimension (when using the definitions from [3] or [2]).The skeleton dimension κ is upper bounded by the max leaf number. Moreover, for anygraph on at least 3 vertices there are edge weights for which both parameters are equal. Asthe max leaf number is an upper bound for the pathwidth pw , it follows that κ ≥ pw . Thisimproves a result of Blum and Storandt, who showed that one can choose edge weightsfor any graph such that the skeleton dimension is at least ( pw − / (log | V | + 2) [11].The resulting parameter hierarchy is illustrated in Figure 1. In the second part of the paperwe show hardness for two problems in transportation networks.We show that computing the highway dimension is NP-hard when using the most recentdefinition from [1]. This answers an open question stated in [18], where NP-hardness wasonly shown for the definitions used in [3] and [2].We study the k -Center problem in graphs of low skeleton dimension. We extend aresult from [17] and show how graphs of low doubling dimension can be embedded intographs of low skeleton dimension. It follows that for any (cid:15) > − (cid:15) )-approximation on graphs of skeleton dimension O (log | V | ). We consider undirected graphs G = ( V, E ) and denote the number of nodes and edgesby n and m , respectively. Let ∆ be the maximum degree of G . For weighted graphs, let ‘ : E → Q + be the cost function. For nodes u, v ∈ V , let dist G ( u, v ) (or simply dist( u, v ))be length of the shortest path from u to v in G . A weighted graph G = ( V, E ) is metric if( V, dist G ) is a metric, i.e. its edge weights satisfy the triangle inequality, that is for all nodes u, v, w ∈ V we have dist( u, w ) ≤ dist( u, v ) + dist( v, w ). We assume that the shortest pathbetween any two nodes of G is unique, which can be achieved e.g. by slightly perturbing theedge weights. For u ∈ V and r ∈ R , we define the ball around the node u of radius r as B r ( u ) = { v ∈ V | dist( u, v ) ≤ r } . The length of a path π is denoted by | π | . The skeleton dimension was introduced by Kosowski and Viennot to analyze the performanceof hub labels, a route planning technique used for road networks [25]. To define it formally,we first need to introduce the geometric realization ˜ G = ( ˜ V, ˜ E ) of a graph G = ( V, E ) withedge weights ‘ . Intuitively, ˜ G is a continuous version of G , where every edge is subdividedinto infinitely many infinitely short edges. This means that V ⊆ ˜ V , for all u, v ∈ V we havedist ˜ G ( u, v ) = dist G ( u, v ) and for every edge { u, v } of G and every 0 ≤ α ≤ ‘ ( { u, v } ) there isa node w ∈ ˜ V satisfying dist( u, w ) = α and dist( w, v ) = ‘ ( { u, v } ) − α .For a node s ∈ V let T s be the shortest path tree of s and let ˜ T s be its geometricrealization. Recall that shortest paths are unique, and hence the same holds for T s and ˜ T s .The skeleton T ∗ s is defined as the subtree of ˜ T s induced by the nodes v ∈ ˜ V that have adescendant w in ˜ T s satisfying dist( v, w ) ≥ / · dist( s, v ). Intuitively, we obtain T ∗ s by takingevery shortest path with source s , cutting off the last third of the path and taking the unionof the truncated paths. For a radius r ∈ R let Cut rs be the set of all nodes u in T ∗ s satisfyingdist( s, u ) = r . (cid:73) Definition 1 (Skeleton Dimension) . The skeleton dimension κ of a graph G is max s,r | Cut rs | . Intuitively, a graph has low skeleton dimension, if for any starting node s there are onlya few main roads that contain the major central part of ever shortest path originating from Hierarchy of Transportation Network Parameters and Hardness Results
Max Leaf
HighwayDimension 1Distance toLinear Forest
Bandwidth
SkeletonDimension HighwayDimension 2
Pathwidth
MaximumDegree
Treewidth h -index AcyclicChromatic strict boundgeneral boundincomparable (a)
Relationships between general graph parameters. ∆ ddim κ hd hd hd f hd hd ( hd + 1)2 hd (∆ + 1) hd f hd (∆ + 1) hd hd ( hd + 1) (b) Relationships between maximum degree ∆, doubling dimension ddim , skeleton dimension κ anddifferent highway dimensions. Figure 1
Relationships between graph parameters. New results are highlighted in green. Solidlines denote strict bounds (e.g. treewidth ≤ pathwidth), dashed lines denote general bounds (e.g.pathwidth ≤ distance to linear forest + 1). Dotted lines denote incomparabilities. . Blum 5 s . Clearly, the skeleton dimension can be computed in polynomial time by computing theshortest path tree and its skeleton for every node s ∈ V and determining Cut rs for everyradius r ∈ R . On large networks, a naïve implementation is still impracticable, but in [11] itwas shown that it is possible to compute κ even for networks with millions of vertices.Related to the skeleton dimension is the doubling dimension, which was introduced as ageneralization of several kinds of metrics, e.g. Euclidean or Manhattan metrics. (cid:73) Definition 2 (Doubling Dimension) . A graph G is d -doubling, if for any radius r , any ballof radius r is contained in the union of d balls of radius r / . If d is the smallest such integer,the doubling dimension of G is log d . Computing the doubling dimension is NP-hard [21]. Kosowski and Viennot showed thata graph with skeleton dimension κ is (2 κ + 1) doubling [25]. The highway dimension was introduced by Abraham et al., motivated by the observation ofBast et al. that in road networks, all shortest paths leaving a certain region pass throughone of a small number of nodes [7, 8]. In the literature, several slightly different definitionsof the highway dimension can be found. The first one was given in [3]. (cid:73)
Definition 3 (Highway Dimension 1) . The highway dimension of a graph G is the smallestinteger hd such that for any radius r and any node u there is a hitting set S ⊆ B r ( u ) ofsize hd for the set of all shortest paths π satisfying | π | > r and π ⊆ B r ( u ) . In [19, 20], a generalized version of hd was used, where balls of radius c · r for c ≥ c and highway dimension ofΩ( n ) w.r.t. radius c > c .In [2] the highway dimension was defined as follows. (cid:73) Definition 4 (Highway Dimension 2) . The highway dimension of a graph G is the smallestinteger hd such that for any radius r and any node u there is a hitting set S ⊆ V of size hd for the set of all shortest paths π satisfying r ≥ | π | > r that intersect B r ( u ) . The definition of hd requires to hit all shortest paths contained in the ball of radius4 r , while for hd only the shortest paths intersecting the ball of radius 2 r need to be hit.Hence, we have hd ≤ hd . Abraham et al. motivate their new definition with the fact thata smaller highway dimension can be achieved on real-world instances, while previous resultsstill hold [2]. Both previously defined highway dimensions are incomparable to the maximumdegree and the doubling dimension [3].In [1], a continuous version of the highway dimension hd was introduced, which is basedon the geometric realization. For the definition, assume w.l.o.g. that ‘ ( e ) ≥ e ∈ E . (cid:73) Definition 5 (Continuous Highway Dimension) . The continuous highway dimension of agraph G is the smallest integer f hd such that for any radius r ≥ and any node u ∈ ˜ V ofthe geometric realization ˜ G there is a hitting set S ⊆ V of size f hd for the set of all shortestpaths π satisfying r ≥ | π | > r that intersect B r ( u ) . Clearly, we have hd ≤ f hd . In [25] it was observed that f hd is upper bounded by(∆ + 1) hd . Along the lines of Definition 3, we can also introduce the continuous version f hd Hierarchy of Transportation Network Parameters and Hardness Results of hd . It holds that hd ≤ f hd ≤ (∆ + 1) hd and moreover f hd ≤ f hd . In [1], yet anotherdefinition of the highway dimension was given. It is based on the notion of r -significantshortest paths. (cid:73) Definition 6 ( r -significant shortest path) . For r ∈ R , a shortest path π = v . . . v k is r -significant iff it has an r -witness path π , which means that π is a shortest path satisfying | π | > r and one of the following conditions hold: (i) π = π , or (ii) π = v π , or π = πv k +1 ,or (iv) π = v πv k +1 for nodes v , v k +1 ∈ V . In other words, π is r -significant, if by adding at most one vertex to every end we canobtain a shortest path π of length more than r (the r -witness). For r, d ∈ R , a shortest path π is ( r, d )-close to a vertex v , if there is an r -witness path π of π that intersects the ball B d ( v ). (cid:73) Definition 7 (Highway Dimension 3) . The highway dimension of a graph G is the smallestinteger hd such that for any radius r and any node u there is a hitting set S ⊆ V of size hd for the set of all shortest paths π that are ( r, r ) -close to u . The advantage of the latest definition is that it also captures continuous graphs. Inparticular, it was shown that hd ≤ f hd ≤ hd [1]. Hence there is no need for a continuousversion of hd , apart from the fact that there is no meaningful notion of an r -witness in acontinuous graph.It can be easily seen that hd ≤ hd as every shortest path π that is longer than r andintersects B r ( u ) is also ( r, r )-close to u (using π itself as the r -witness). Moreover, theskeleton dimension κ is a lower bound for hd , i.e. κ ≤ hd [25]. Feldmann et al. showed that hd ≤ hd ( hd + 1) [18]. Combining their proof with [1] yields that f hd ≤ hd ( hd + 1).Computing the highway dimensions hd and hd is NP-hard [18]. In Section 4.1 we showthat this also holds for hd , which answers an open question stated in [18]. We now provide an overview of several classic graph parameters. They are all defined onunweighted graphs, but we can also apply them to weighted graphs, simply neglecting edgeweights. We start with introducing the treewidth and the related parameters pathwidth andbandwidth. (cid:73)
Definition 8 (Treewidth) . A tree decomposition of a graph G = ( V, E ) is a tree T = ( X , E ) where every node (also called bag ) X ∈ X is a subset of V and the following properties aresatisfied: (i) S X ∈X X = V , (ii) for every edge { u, v } ∈ E there is a bag X ∈ X containingboth u and v , and (iii) for every u ∈ V , the set of all bags containing u induce a connectedsubtree of T . The width of a tree decomposition T = ( X , E ) is the size of the largest bagminus one, i.e. max X ∈X ( | X | − . The treewidth tw of a graph G = ( V, E ) is defined as theminimum width of all tree decompositions of G . (cid:73) Definition 9 (Pathwidth) . A path decomposition of a graph G is a tree decomposition of G that is a path. The pathwidth pw of G is the minimum width of all path decompositions of G . It follows directly from the definitions, that the pathwidth is an upper bound for thetreewidth and one can show that the minimum degree is a lower bound for the treewidth [27].The maximum degree ∆ is incomparable to both treewidth and pathwidth, as for a squaregrid graph we have ∆ = 4 and tw ∈ Ω( √ n ) whereas for a star graph we obtain ∆ ∈ Ω( n )and pw = 1. . Blum 7 (cid:73) Definition 10 (Bandwidth) . A vertex labeling of a graph G = ( V, E ) is a bijection f : V →{ , . . . , n } . The bandwidth of G is the minimum of max {| f ( u ) − f ( v ) | : { u, v } ∈ E } , takenover all vertex labelings f of G . It was shown that the bandwidth bw is a tight upper bound for the pathwidth [24], andthat ∆ ≤ · bw [27]. (cid:73) Definition 11 (Max Leaf Number) . The max leaf number ml of a graph G is the maximumnumber of leaves of all spanning trees of G . (cid:73) Definition 12 (Distance to Linear Forest) . The distance to linear forest (also known asdistance to union of paths) of a graph G = ( V, E ) is the size of the smallest set S ⊆ V thatseparates G into a set of disjoint paths. (cid:73) Definition 13 ( h -Index) . The h -index of a graph G = ( V, E ) is the largest integer h suchthat G has h vertices of degree at least h . The max leaf number is closely related to the notion of a connected dominating set. It isan upper bound for several graph parameters. It was shown that for the max leaf number ml and the distance to linear forest dl we have dl ≤ ml − dl andpathwidth pw it is known that pw ≤ dl + 1 [13]. Clearly, the h -index is a lower bound forthe maximum degree. It was shown that the h -index is incomparable to the treewidth [27]. In this section we show relationships between skeleton dimension, highway dimension andother graph parameters. We will see that the max leaf number is an upper bound for theskeleton dimension and the bandwidth, whereas many of the remaining parameters arepairwise incomparable. This shows that they are all useful and worth studying.
We first relate the max leaf number to the skeleton dimension and the bandwidth. We willuse the fact, that every tree has as least as many leaves as any subtree. (cid:73)
Observation 14.
Let T be a subtree of a tree T and let L and L be the leaves of T and T , respectively. Then we have | L | ≤ | L | . This allows to show that the max leaf number is an upper bound for the skeletondimension. (cid:73)
Theorem 15.
For the skeleton dimension κ and the max leaf number ml we have κ ≤ ml .For any unweighted undirected graph on n ≥ nodes there are metric edge weights such that κ = ml . Proof.
Let G = ( V, E ) be a graph. Consider the skeleton T ∗ s of some node s ∈ V that hasa cut C of size κ . As for any two distinct nodes u, v ∈ C the lowest common ancestor in T ∗ s is distinct from u and v , T ∗ s has at least κ leaves. The skeleton T ∗ s is a subtree of theshortest path tree T s of s , so Observation 14 implies that T s has at least κ leaves. As T s is aspanning tree of G it follows that κ ≤ ml . Hierarchy of Transportation Network Parameters and Hardness Results
To show that the bound is tight, consider a spanning tree T = ( V, E T ) of an unweightedgraph G = ( V, E ) with ml leaves. We choose edge weights ‘ such that the skeleton dimensionof the resulting weighted graph equals ml . Let ‘ ( { u, v } ) = { u, v } ∈ E T and u or v is a leaf of T / n if { u, v } ∈ E T and neither u nor v is a leaf of T s of T .Such a node exists if n >
2. We observe that the shortest path tree T s of s is equal to T as for any vertex v we have dist( s, v ) <
3, and hence no edge e ∈ E \ E T can be containedin T s . Moreover, for any leaf v we have dist( s, v ) ≥ v we havedist( s, v ) <
1. Consider now the skeleton T ∗ s . Any leaf of T ∗ s has distance at least / · > s . As T ∗ s has ml leaves, the cut of T ∗ s at radius / has size ml .Note that in general, the resulting graph is not metric. To fix this, let dist T ( u, v ) be theshortest path distance from u to v when applying the previously chosen edge weights. For { u, v } ∈ E T we define ‘ as previously, but for { u, v } 6∈ E T choose ‘ ( u, v ) = dist T ( u, v ) − (cid:15) where for every edge, (cid:15) is chosen from (0 , / n ) such that shortest paths are unique. Consideran internal node s of T . The shortest path tree T s of s may now differ from T , but thenumber of leaves of T s is still ml . For any leaf v of T we have now dist( s, v ) > − n / n ≥ / and for any internal node v we have dist( s, v ) <
1. Hence, the cut of T ∗ s at radius 1 has size ml . (cid:74) As the max leaf number ml is an upper bound for the pathwidth pw , it follows thatfor any graph G on n ≥ κ ≥ pw . This improvesa result of Blum and Storandt, who showed that there are edge weights such that κ ≥ ( pw − / (log n + 2) [11].Sorge et al. showed that the bandwidth can be upper bounded by two times the max leafnumber [27]. We slightly modify their proof to remove the factor of 2 and show that theresulting bound is tight. (cid:73) Lemma 16.
For the max leaf number ml and the bandwidth bw we have bw ≤ ml . Thisbound is tight. Proof.
Let T be a BFS tree of a graph G = ( V, E ) and let f : V → { , . . . , n } be a vertexlabeling that assigns to every node the time of its BFS discovery. W.l.o.g. we assumethat f ( v i ) = i . Choose an edge { v i , v j } ∈ E maximizing f ( v j ) − f ( v i ). It follows that bw ≤ f ( v j ) − f ( v i ) = j − i .Observe that in the BFS tree T , the node v i is the parent of v j as by the choice of { v i , v j } there is no k < i such that { v k , v j } ∈ E . Consider the subtree T of T induced by the nodes { v , . . . , v j } . As v i is the parent of v j and nodes are ordered by their discovery time, itfollows that v i +1 , . . . , v j are leaves of T . Observation 14 implies T has at least ( j − i ) leaves.Tightness follows from the complete graph K n where bw = ml = n − (cid:74) We now show incomparabilities between several parameters, which means that they are allworth studying. In [27] it was proven that the treewidth is incomparable to the h -index. Weobserve that the same holds for the pathwidth. (cid:73) Theorem 17.
The pathwidth and h -index are incomparable. . Blum 9 Proof.
The √ n × √ n grid graph has pathwidth √ n and h -index at most 4. The caterpillartree with d backbone vertices of degree d has pathwidth 1 and h -index d . (cid:74) We proceed with relating the highway dimensions hd and hd to the treewidth andpathwidth. In [19] it was observed that graphs of low highway dimension hd do nothave bounded treewidth, as the complete graph on vertex set { , . . . , n } with edge weights ‘ ( { i, j } ) = 4 max( i,j ) has highway dimension hd = 1 and treewidth n − The completegraph K n has indeed a minimum degree of n −
1, which is a lower bound for the treewidth.On the other hand, there are graphs of constant bandwidth and a linear highway dimension hd . For instance, consider a complete caterpillar tree on b backbone vertices of degree 3.Its bandwidth is 2. Choose the weight of an edge as / n if it is a backbone edge and as 1otherwise. Every edge of weight 1 is a shortest path intersecting the ball of radius 1 aroundsome fixed backbone vertex and hence hd ≥ b = n / −
2. This gives us the follows theorem. (cid:73)
Theorem 18.
The highway dimensions hd and hd are incomparable to the bandwidthand the minimum degree. We would also like to relate the skeleton dimension to bandwidth and treewidth. Ongeneral graphs, it is easy to show, that the skeleton dimension is incomparable to the othertwo parameters. For instance, a star graph has treewidth 1 and linear skeleton dimension,whereas a complete graph has linear treewidth, but we can choose edge weights such that theshortest path tree of every vertex becomes a path which implies a constant skeleton dimension.However, by choosing such weights for the latter graph, most edges become useless as theydo not represent a shortest path and removing all unnecessary edges produces a graph oflow treewidth. Still, we can show, that even on metric graphs the skeleton dimension isincomparable to both bandwidth and treewidth. (cid:73)
Theorem 19.
On metric graphs the skeleton dimension and the bandwidth are incomparable.
Proof.
Consider the complete caterpillar tree on b backbone vertices of degree 3. It has abandwidth of 2. Set the weight of every backbone edge to 1 and pick an arbitrary backbonevertex v . For the remaining edges, choose edge weights such that all leaves have the samedistance d ≥ v . It follows that the skeleton dimension of the weighted caterpillar treeequals the number of leaves which is b + 2 = n / + 1.The complete binary tree B d +1 of depth 2 d + 1 has pathwidth d [14]. We show thatthere are edge weights for B d +1 such that the skeleton dimension is at most 3. Let s bethe root of B d +1 . We call the depth of a vertex in the tree also its level and choose theweight of an edge { v, w } as ‘ ( { v, w } ) = 3 − j if v and w are level j and level ( j + 1) vertices,respectively.Let v be a level i vertex. We show that for any radius r we have | Cut rv | ≤
3. Clearly theshortest path π form v to s is contained in the skeleton T ∗ v of the shortest path tree T v as theroot s has a descendant w satisfying dist( s, w ) ≥ / · dist( v, s ). For 0 ≤ j ≤ i let v j be theunique level j vertex on the path π and let w be a descendant of some v j such that the shortestpath from v j to w is edge-disjoint from π . Assume that the vertex w is contained in theskeleton T ∗ v . This means that w has a descendant w such that dist( w, w ) ≥ / · dist( v, w ).As moreover dist( v j , w ) = dist( v j , w ) − dist( w, w ) and dist( v j , w ) ≤ dist( v, w ), it follows that The edge weights chosen in [19] are actually ‘ ( { i, j } ) = 4 min( i,j ) , which results in a non-metric graph.Removing all edges that are not a shortest path yields a star graph of treewidth 1. / · dist( v j , w ) ≤ dist( v j , w ) which implies dist( v j , w ) ≤ / P d +1 x = j − x < / P ∞ x = j − x = · − j +1 = 3 − j .To bound the size of Cut rv , consider a radius r > y be the node in shortest pathfrom v to the root s that maximizes dist( v, y ) while satisfying r := dist( v, y ) ≤ r . Let j bethe level of y . From our previous observation it follows that y is the only vertex that hasdistance r from v and is contained in the skeleton T ∗ v . Moreover, there are at most threevertices at distance r − r from y . It follows that | Cut rv | ≤ B d +1 has skeletondimension κ ≤ (cid:74)(cid:73) Theorem 20.
On metric graphs the skeleton dimension and the treewidth are incomparable.
Proof.
The star graph S n on n vertices has treewidth 1 and skeleton dimension n − √ n ) and constant skeleton dimension. Considera square grid graph G on the vertex set V = { v , . . . , v n } . Subdivide every edge { u, v } byinserting two vertices x uv and y uv , i.e. replace the edge { u, v } through a path u x uv y uv v .Connect the vertices v , . . . , v n through a path P and denote the resulting graph by G =( V , E ). The original grid graph G has treewidth √ n and is a minor of G . Hence, G hastreewidth Ω( √ n ).We now choose edges weights for G resulting in a constant skeleton dimension. For everyedge e that is part of the path P , let ‘ ( e ) = 1. Consider an edge { u, v } of G that was replacedby the path u x uv y uv v and denote the shortest path distance between u and v on the path P by dist P ( u, v ). We choose ‘ ( { u, x uv } ) = ‘ ( { y uv , v } ) = 1 and ‘ ( { x uv , y uv } ) = dist P ( u, v ) + / .It is easy to verify that the resulting graph is metric.To bound the skeleton dimension, we use the following claim: For every edge { u, v } of G ,neither of the shortest paths from u to x uv or from v to y uv contains the edge { x uv , y uv } . Toprove the claim, observe that by concatenating the subpath of P between u and v and theedge { v, y uv } , we obtain a path of length dist P ( u, v ) + 1. Any path from u to y uv containingthe edge { x uv , y uv } has length dist P ( u, v ) + / . The case of v and x uv is symmetric.It follows that in G the shortest path tree of a vertex s cannot contain the edge { x uv , y uv } unless s ∈ { x uv , y uv } , as any subpath of a shortest path must be a shortest pathitself. Consider the shortest path tree T s of some vertex s ∈ V . The previous claim impliesthat T s is a caterpillar tree where P is the backbone path. Moreover, T s ha maximum degree∆ ≤ r > rs . For r ≤
1, the setCut rs intersects only edges incident to s and hence | Cut rs | ≤
6. For 1 < r ≤
2, the set Cut rs intersects only edges incident to the two neighbors of s on P , which implies | Cut rs | ≤ r > | Cut rs | ≤ v ˜ P , the distance to itsfurthest descendant is less than 1 < r/ rs intersects only edges fromthe path P . Similarly, it can be shown that | Cut rs | ≤ s V (i.e. s = x uv or s = y uv ). Itfollows that the skeleton dimension of G is κ ≤ (cid:74) So far, it was only known that there can be an exponential gap between skeleton andhighway dimension [25]. However, we can use the graph G from the previous proof to showthat the skeleton dimension and the highway dimensions hd and hd are incomparable.Let { v , , . . . , v q,q } be the vertex set of the original grid graph and choose the path P usedin the construction of G as v , . . . v ,q v , . . . v ,q . . . v q, . . . v q,q . In the resulting graph G , for i ∈ { , . . . , q } , the shortest path from v ,i to v ,i has length q and hence the edge e i = { x v ,i ,v ,i , y v ,i ,v ,i } has length q + . As any edge of { e , . . . , e q } intersects the ballaround v , of radius 2 q and no two of this edges share a common vertex, the highwaydimension hd of G is at least q = √ n . The star graph on n vertices with unit edge weights . Blum 11 has a skeleton dimension of n − hd of 1, so we obtain thefollowing corollary. (cid:73) Corollary 21.
The skeleton dimension is incomparable to both highway dimensions hd and hd . Finally it can be shown that the distance to linear forest dl is incomparable to thebandwidth bw , the skeleton dimension κ and the highway dimensions hd and hd . Forinstance, a caterpillar tree of constant maximum degree has a distance to linear forest of Ω( n ),but constant bandwidth, skeleton dimension and highway dimensions (for suitably chosen edgeweights), whereas there are star-like graphs for which dl ∈ O (1) and bw, κ, hd , hd ∈ Ω( n ). (cid:73) Theorem 22.
The distance to linear forest is incomparable to the bandwidth, the skeletondimension and the highway dimensions hd and hd . Proof.
We will use the fact that the caterpillar tree C b on b backbone vertices of degree 3has a distance to linear forest of b = n / − Bandwidth.
The caterpillar C b has bandwidth 2. The star graph S n on n vertices has abandwidth of b n / c and a distance to linear forest of 1. Skeleton dimension.
Consider the caterpillar C b and choose the weight of an edge { u, v } as 2 if u and v are both backbone vertices and as 1 otherwise. The skeleton T ∗ s of anyvertex s contains exactly one vertex of degree 3 (the backbone vertex that is closest to s )and no vertex of degree more than 3. Hence, the skeleton dimension is 3. The star graph S n on n vertices with unit edge weights has a skeleton dimension of n − Highway dimensions.
Consider the caterpillar C b and choose the weight of an edge { u, v } as 5 if u and v are both backbone vertices and as 1 otherwise. To bound the highwaydimension hd , consider some node v and let r >
0. Consider a maximum path P ⊆ B r ( v )containing only backbone vertices. It holds that | P | ≤ r . We can greedily choose a set S ⊆ P such that | S | ≤ π of P of length | π | ≥ r − S . Considera path π ⊆ B r ( v ) that is not hit by S . The path π contains at most two edges of length 1incident to a leaf and a subpath of P that has length less than r −
2. Hence, π has length atmost r . It follows that for any v ∈ V and any r > π satisfying | π | > r and π ⊆ B r ( v ) with at most 7 vertices, which means that hd ≤ l leaves, subdivide every edge by inserting one vertex and choosethe weight of every edge in the resulting graph as 1. We obtain a graph of distance to linearforest 1 and highway dimension hd = l , as every edge incident to a leaf is a shortest path oflength 1 intersecting the ball of radius 1 around the central vertex. (cid:74) In this section we show hardness for two problems in transportation networks. We firstshow that computing the highway dimension in NP-hard, even when using the most recentdefinition. Then we consider the k -Center problem and show that for any (cid:15) >
0, computinga (2 − (cid:15) )-approximation is NP-hard on graphs of skeleton dimension O (log n ). In [18] it was shown that computing the highway dimension hd is NP-hard. The presentedreduction is from Vertex Cover and also works for hd . It does not directly carry over to hd as the constructed graph has maximum degree ∆ = n − hd ≥ ∆. Still,using a slightly different reduction, we can show NP-hardness for the computation of hd . (cid:73) Theorem 23.
Computing the highway dimension hd is NP-hard. Proof.
We present a reduction from
Vertex Cover on graphs with maximum degree ∆ ≤ G = ( V, E ) on n nodes satisfying ∆ ≤
3. We construct a weightedgraph G = ( V , E ) as follows. Add a single node x and for any node v ∈ V , add a new node v ∗ and the edges { v, v ∗ } and { v ∗ , x } . For an edge e ∈ E choose edge weight ‘ ( e ) = 5 if e isincident to x and ‘ ( e ) = 1 otherwise.Let C be a minimum vertex cover of G . We may assume that | C | > ((∆+1) − / ∆ ∈ O (1)as for any constant c we can decide in polynomial time whether G has a minimum vertexcover of size c . We show that G has highway dimension hd = | C | + n + 1. Observe that hd is still linear in n , but it may vary between n + 1 and 2 n , depending on | C | .Let 0 < r < / . Consider a node u ∈ V . Let N be the closed neighborhood of theball around u of radius 2 r , i.e. v ∈ N iff v ∈ B r ( u ) or v is adjacent to a node w ∈ B r ( u ).Clearly, N is a hitting set for all shortest paths that are ( r, r )-close to u . For u = x , the ball B r ( u ) contains at most P i =0 (∆ + 1) i nodes, as 2 r <
5. Moreover, every node in B r ( u ) hasat most ∆ + 1 neighbors. Hence, N is a hitting set of size P i =0 (∆ + 1) i = ((∆ + 1) − / ∆for all shortest paths that are ( r, r )-close to u . For u = x , we have N = V \ V and therefore | N | = n + 1.Let r = / . The ball around x of radius 2 r = 5 is B r ( x ) = V \ V . Any edge { u, v } ∈ E is ( r, r )-close to x , as u ∗ u v v ∗ is an r -witness. Moreover, any node u ∈ V \ V is a shortestpath that is ( r, r )-close to x . However, a single node u ∈ V is not r -significant, as it can onlybe extended to a witness of length 2 < r . Hence, a shortest path π is ( r, r )-close to x iff andonly if π ∈ E or π ∈ V \ V . Consider a smallest hitting set H ⊆ V for all shortest paths thatare ( r, r )-close to x . We have ( V \ V ) ⊆ H , we H needs to hit all paths that consist of onesingle node v ∈ V \ V . Moreover, H needs to hit all edges E . In other words, H consists of V \ V and a vertex cover for G . Hence, the hitting set H has size | V \ V | + | C | = | C | + n + 1.Observe that for r = / , any r -significant shortest path in G is ( r, r )-close to x , as anynode of G has a neighbor contained in B r ( x ). Hence, for any node u ∈ V , there is a hittingset for all shortest paths that are ( r, r )-close to u of size at most | C | + n + 1. Moreover,for any node u and any r > / , a shortest path can only be ( r, r )-close to u , if it is also( / , u . Hence, for any u ∈ V and any r > / , for all shortest paths that are( r, r )-close to u there is a hitting set of size at most | C | + n + 1.We conclude that the highway dimension of G is hd = | C | + n + 1 if and only if G hasa minimum vertex cover of size | C | . (cid:74) k -Center In the k -Center problem, we are given a graph G = ( V, E ) with positive edge weights andthe goal is to select k center nodes C ⊆ V while minimizing max u ∈ V min v ∈ C dist( u, v ), thatis the maximum distance from any node to the closest center node. A possible scenario isthat one wants to place a limited number of hospitals on a map such that the maximumdistance from any point to the closest hospital is minimized.We will prove that computing a (2 − (cid:15) )-approximation on graphs with low skeletondimension is NP-hard. For that purpose, we first show the following lemma, which is anon-trivial extension of a result of Feldmann [17]. The aspect ratio of a metric ( X, dist X )is the ratio of the maximum distance between any pair of vertices in X and the minimumdistance. . Blum 13 (cid:73) Lemma 24.
Let ( X, dist X ) be a metric of constant doubling dimension d and aspect ratio α . For any < (cid:15) < it is possible to compute a graph G = ( X, E ) in polynomial time thathas the following properties: (a) for all u, v ∈ X we have dist X ( u, v ) ≤ dist G ( u, v ) ≤ (1 + (cid:15) ) dist X ( u, v ) , (b) the graph G has highway dimension hd ∈ O ((log( α ) /(cid:15) ) d ) , and (c) the graph G has skeleton dimension κ ∈ O ((log( α ) /(cid:15) ) d ) , Proof.
In [17] it was shown, how to compute a Graph H that satisfies properties a and b.This was done by choosing so called hub sets Y i ⊆ X for all i = 0 , , . . . , L = d log α e suchthat in H any shortest path in the range (2 i , i +1 ] contains some node from Y i . Moreover,for any vertex u ∈ X and any i there is a hub v ∈ Y i satisfying dist X ( u, v ) ≤ (cid:15) i − (1+ (cid:15) ) L and forany two distinct hubs u, v ∈ Y i we have dist X ( u, v ) > (cid:15) i − (1+ (cid:15) ) L . The hub sets form a hierarchy,i.e. Y i ⊇ Y j for all i < j . In the computed graph H , there is an edge between two vertices u and v of length (1 + (cid:15) (1 − i/L )) dist X ( u, v ) if and only if i = max { j | { u, v } ⊆ Y j } . We callsuch an edge also a level i edge . Moreover, Y = X is chosen and hence the graph H has (cid:0) | X | (cid:1) edges, which implies a maximum degree of ( | X | − i edges { u, v } of H satisfyingdist X ( u, v ) > i +1 , which yields a graph G . The following claim shows that this does notaffect the shortest path structure of the graph. (cid:66) Claim 25.
The graph G fulfils properties a and b. Proof.
We show that the constructed graph G has exactly the same shortest paths as H .This implies that G fulfils properties a and b. Consider an edge { u, v } that was removedfrom H . We claim that { u, v } is longer than the shortest path from u to v . As the edge wasremoved, we have dist X ( u, v ) > i +1 where i = max { j | { u, v } ⊆ Y j } is the level of { u, v } .The length of { u, v } in H is d uv = (1 + (cid:15) (1 − i/L )) dist X ( u, v ). Any shortest path longerthan 2 i +1 contains some hub from Y i +1 . Hence, if u, v Y i +1 , the edge { u, v } cannot be ashortest path in H and we are done. Assume now that u ∈ Y i +1 . This implies v ∈ Y i \ Y i +1 as { u, v } has level i . A property of the hub set Y i +1 is that there is a hub w ∈ Y i +1 satisfyingdist X ( v, w ) ≤ (cid:15) i − (1+ (cid:15) ) L .As u, w ∈ Y i +1 , the edge { u, w } has in H length at most d uw = (cid:18) (cid:15) (cid:18) − i + 1 L (cid:19)(cid:19) dist X ( u, w ) ≤ (cid:18) (cid:15) (cid:18) − i + 1 L (cid:19)(cid:19) (dist X ( u, v ) + dist X ( v, w )) ≤ (cid:18) (cid:15) (cid:18) − i + 1 L (cid:19)(cid:19) dist X ( u, v ) + 2 · dist X ( v, w ) . This means, that H contains a path from u to w of length at most d uw . Moreover, property aimplies that H contains a v - w -path of length at most d wv = 2 · dist X ( v, w ). It follows thatby concatenating the shortest paths from u to w and from w to v , we obtain a u - v -path whose length is upper bounded by d uw + d wv ≤ (cid:18) (cid:15) (cid:18) − i + 1 L (cid:19)(cid:19) dist X ( u, v ) + 2 · dist X ( v, w ) + 2 · dist X ( v, w )= (cid:18) (cid:15) (cid:18) − iL (cid:19)(cid:19) dist X ( u, v ) − (cid:15)L · dist X ( u, v ) + 4 · dist X ( v, w ) ≤ d uv − (cid:15)L · dist X ( u, v ) + 4 · dist X ( v, w ) < d uv − (cid:15)L i +1 + 4 · (cid:15) i − (1 + (cid:15) ) L = d uv − (cid:15)L · (cid:18) i +1 − i +1 (1 + (cid:15) ) (cid:19) < d uv . Hence, the edge { u, v } is longer than the shortest path from u to v . (cid:67) It follows that if { u, v } is a long edge in G , then both u and v must be important hubs. (cid:66) Claim 26. In G , for any edge { u, v } of length more than 2 i we have u, v ∈ Y i − . Proof.
Consider an edge { u, v } of level j ≤ i −
2. As { u, v } was not removed from H , wehave dist X ( u, v ) ≤ j +1 ≤ i − . An upper bound of 2 i on the length of { u, v } follows, as thelength of { u, v } was chosen as (1 + (cid:15) (1 − j/L )) · dist X ( u, v ) < · dist X ( u, v ) ≤ i . (cid:67) It remains to bound the skeleton dimension of G . For some vertex s and radius r considerthe set Cut rs in the skeleton of s at radius r and choose some vertex v ∈ Cut rs . Let w be afurthest descendant of v in the shortest path tree of s and choose i such that for the distance r = dist G ∗ ( v, w ) we have 2 i < r ≤ i +1 . As v is contained in the skeleton, it follows thatfor the distance r = dist G ∗ ( s, v ) we have r ≤ r ≤ i +2 .Choose an edge { ( v, * v } of G satisfying dist G ∗ ( ( v, * v ) = dist G ∗ ( ( v, v ) + dist G ∗ ( v, * v ). In otherwords, ( v and * v are the parent and child node of v when considering only nodes from the(discrete) graph G . We claim that the vertex v has a descendant in the shortest path tree of s which is contained in the hub set Y i − . To prove this, observe that dist G ( ( v, w ) ≥ r > i .This implies that (i) dist G ( ( v, * v ) > i − or (ii) dist G ( * v, w ) > i − . Consider case (i). As theedge { ( v, * v } is contained in G and has length more than 2 i − , it follows from Claim 26 that * v ∈ Y i − . In case (ii), it follows from dist G ( * v, w ) > i − that the shortest path from * v to w muss pass through a hub from Y i − ⊆ Y i − .Denote the ball B r ( s ) in G simply by B r . It holds that dist G ( s, w ) = r + r < i +3 . Thismeans that every vertex v ∈ Cut rs has a descendant v in the shortest path tree which iscontained in Y i − ∩ B i +3 . As the skeleton of s is a tree, for all distinct vertices u, v ∈ Cut rs we have u = v . Hence, we have | Cut rs | ≤ | Y i − ∩ B i +3 | .It was shown that if ( X, dist X ) is a metric with doubling dimension d , for any subset X ⊆ X of aspect ratio β , the size of X is bounded by 2 d d log β e ≤ (2 β ) d [22]. As thediameter of the ball B i +3 is at most 2 i +4 (which according to property a also bounds thediameter of the ball w.r.t. the metric ( X, dist X )) and any two distinct hubs u, v ∈ Y i − have distance dist X ( u, v ) > (cid:15) i − (1+ (cid:15) ) L , the aspect ratio of Y i − ∩ B i +3 w.r.t. dist X is β < i +4 / (cid:15) i − (1+ (cid:15) ) L = 2 (1 + (cid:15) ) L/(cid:15) . It follows that | Y i − ∩ B i +3 | ≤ (2 · (1 + (cid:15) ) L/(cid:15) ) d . As (cid:15) < rs is bounded by (2 L/(cid:15) ) d and we obtain that the skeleton dimension of G is κ ∈ O (( L/(cid:15) ) d ) = O ((log( α ) /(cid:15) ) d ). (cid:74) . Blum 15 Feldmann [17] observed that due to a result of Feder and Greene [16], it is NP-hard forany (cid:15) > − (cid:15) )-approximation for k -Center on graphs of doubling dimension4 and aspect ratio at most n . Lemma 24 hence implies that it is NP-hard to compute a(2 − (cid:15) )-approximation if the skeleton dimension is in O (log n ). It remains open whether thisalso holds for κ ∈ o (log n ) and in particular for constant skeleton dimension.It was also shown, that under the exponential time hypothesis (ETH) it is not possible tocompute a (2 − (cid:15) )-approximation for k -Center on graphs of highway dimension hd in time2 o ( √ hd · n O (1) [17]. Analogously, Lemma 24 implies a bound of 2 o ( √ κ ) · n O (1) for skeletondimension κ . We summarize our findings in the following theorem. (cid:73) Theorem 27.
For any (cid:15) > , it is NP-hard to compute a (2 − (cid:15) ) -approximation for the k -Center problem on graphs of skeleton dimension κ ∈ O (log n ) . Assuming ETH there isno o ( √ κ ) · n O (1) time algorithm that computes a (2 − (cid:15) ) -approximation. We showed that the skeleton dimension, the highway dimension (when defined as in [3]or [2]) and several other graph parameters are pairwise incomparable. Nevertheless, theskeleton dimension is upper bounded by the max leaf number and lower bounded throughthe maximum degree and the doubling dimension.However, for the highway dimensions hd and hd there are still no tight upper or lowerbounds. Using a grid graph and a complete graph, it can be shown that they are not evencomparable to the minimum degree or the maximum clique size, which are lower bounds for alarge number of graph parameters. Bauer et al. showed, that for any unweighted graph thereare edge weights such that hd ≥ ( pw − / (log / | V | + 2) where pw is the pathwidth [9]. Itremains open whether this bound is tight.It turned out that computing the highway dimension is NP-hard for all three differentdefinitions used in the literature. Still, knowing the highway dimension of real-world networkswill give further insight in the structure of transportation networks and hence it is worthwhileto study whether there are FPT algorithms to compute the highway dimension and to whatextent it can be approximated.We proved that on graphs of skeleton dimension O (log n ) it is not possible to beatthe well-known 2-approximation algorithm by Hochbaum and Shmoys for k -Center . Yet,the experimental results reported in [11] indicate that the skeleton dimension of real-worldnetworks might actually be a constant independent of the size of the network. This raises thequestion whether there is a (2 − (cid:15) )-approximation algorithm for graphs of constant skeletondimension. References Ittai Abraham, Daniel Delling, Amos Fiat, Andrew V. Goldberg, and Renato F. Werneck.Highway dimension and provably efficient shortest path algorithms.
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