Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs
Thomas Bläsius, Philipp Fischbeck, Tobias Friedrich, Maximilian Katzmann
SSolving Vertex Cover in Polynomial Time onHyperbolic Random Graphs
Thomas Bläsius
Hasso Plattner Institute, University of PotsdamPotsdam, [email protected]
Philipp Fischbeck
Hasso Plattner Institute, University of PotsdamPotsdam, Germanyphilipp.fi[email protected]
Tobias Friedrich
Hasso Plattner Institute, University of PotsdamPotsdam, [email protected]
Maximilian Katzmann
Hasso Plattner Institute, University of PotsdamPotsdam, [email protected]
Abstract
The
VertexCover problem is proven to be computationally hard in different ways: It is NP-complete to find an optimal solution and even NP-hard to find an approximation with reasonablefactors. In contrast, recent experiments suggest that on many real-world networks the run timeto solve
VertexCover is way smaller than even the best known FPT-approaches can explain.Similarly, greedy algorithms deliver very good approximations to the optimal solution in practice.We link these observations to two properties that are observed in many real-world networks,namely a heterogeneous degree distribution and high clustering. To formalize these propertiesand explain the observed behavior, we analyze how a branch-and-reduce algorithm performs onhyperbolic random graphs, which have become increasingly popular for modeling real-world networks.In fact, we are able to show that the
VertexCover problem on hyperbolic random graphs can besolved in polynomial time, with high probability.The proof relies on interesting structural properties of hyperbolic random graphs. Since thesepredictions of the model are interesting in their own right, we conducted experiments on real-worldnetworks showing that these properties are also observed in practice. When utilizing the samestructural properties in an adaptive greedy algorithm, further experiments suggest that, on realinstances, this leads to better approximations than the standard greedy approach within reasonabletime.
Theory of computation → Graph algorithms analysis; Theory ofcomputation → Random network models; Mathematics of computing → Random graphs
Keywords and phrases vertex cover, random graphs, hyperbolic geometry, efficient algorithm
Funding
This research was partially funded by the German Research Foundation (DeutscheForschungsgemeinschaft, DFG) – project number 390859508. a r X i v : . [ c s . D S ] F e b Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs
VertexCover is a fundamental NP-complete graph problem. For a given undirectedgraph G on n vertices the goal is to find the smallest vertex subset S , such that each edgein G is incident to at least one vertex in S . Since, by definition, there can be no edge betweentwo vertices outside of S , these remaining vertices form an independent set. Therefore, onecan easily derive a maximal independent set from a minimal vertex cover and vice versa.Due to its NP-completeness there is probably no polynomial time algorithm for solving VertexCover . The best known algorithm for
IndependentSet runs in 1 . n poly( n ) [22].To analyze the complexity of VertexCover on a finer scale, several parameterized solutionshave been proposed. One can determine whether a graph G has a vertex cover of size k byapplying a branch-and-reduce algorithm. The idea is to build a search tree by recursivelyconsidering two possible extensions of the current vertex cover ( branching ), until a vertex coveris found or the size of the current cover exceeds k . Each branching step is followed by a reduce step in which reduction rules are applied to make the considered graph smaller. This branch-and-reduce technique yields a simple O (2 k poly( n )) algorithm, where the exponential portioncomes from the branching. The best known FPT (fixed-parameter tractable) algorithm runsin O (1 . k + kn ) time [7], and unless ETH (exponential time hypothesis) fails, there canbe no 2 o ( k ) poly( n ) algorithm [6].While these FPT approaches promise relatively small running times if the considerednetwork has a small vertex cover, the cover is large for many real-world networks. Nevertheless,it was recently observed that applying a branch-and-reduce technique on real instances is veryefficient [1]. Some of the considered networks had millions of vertices, yet an optimal solution(also containing millions of vertices) was computed within seconds. Most instances were solvedso quickly since the expensive branching was not necessary at all. In fact, the application ofthe reduction rules alone already yielded an optimal solution. Most notably, applying the dominance reduction rule , which eliminates vertices whose neighborhood contains a vertextogether with its neighborhood, reduces the graph to a very small remainder on which thebranching, if necessary, can be done quickly. We trace the effectiveness of the dominance ruleback to two properties that are often observed in real-world networks: a heterogeneous degreedistribution (the network contains many vertices of small degree and few vertices of highdegree) and high clustering (the neighbors of a vertex are likely to be neighbors themselves).We formalize these key properties using hyperbolic random graphs to analyze the perform-ance of the dominance rule. Introduced by Krioukov et al. [17], hyperbolic random graphsare obtained by randomly distributing nodes in the hyperbolic plane and connecting any twothat are geometrically close. The resulting graphs feature a power-law degree distributionand high clustering [14, 17] (the two desired properties) which can be tuned using parametersof the model. Additionally, the generated networks have a small diameter [13]. All of theseproperties have been observed in many real-world networks such as the internet, social net-works, as well as biological networks like protein-protein interaction networks. Furthermore,Boguná, Papadopoulos, and Krioukov showed that the internet can be embedded into thehyperbolic plane such that routing packages between network participants greedily worksvery well [5], indicating that this network naturally fits into the hyperbolic space.By making use of the underlying geometry, we show that VertexCover can be solvedin polynomial time on hyperbolic random graphs, with high probability. This is done byshowing that even a single application of the dominance reduction rule reduces a hyperbolicrandom graph to a remainder with small pathwidth on which
VertexCover can thenbe solved efficiently. Our analysis provides an explanation for why
VertexCover can be . Bläsius, P. Fischbeck, T. Friedrich, M. Katzmann 3 solved efficiently on practical instances. We note that, while our analysis makes use of theunderlying hyperbolic geometry, the algorithm itself is oblivious to it. Besides the runningtime the model predicts certain structural properties that also point us to an adapted greedyalgorithm that is still very efficient and achieves better approximation ratios. We conductedexperiments indicating that these predictions (concerning the structural properties andimproved approximation) actually match the real world for a significant fraction of networks.
Let G = ( V, E ) be an undirected graph. We denote the number of vertices in G with n . The neighborhood of a vertex v is defined as N ( v ) = { w ∈ V | { v, w } ∈ E } and the size of theneighborhood, called the degree of v , is denoted by deg( v ). For a subset S ⊆ V , we use G [ S ]to denote the induced subgraph of G obtained by removing all vertices in V \ S . Furthermore,we use the shorthand notation G ≤ d to denote G [ { v ∈ V | deg( v ) ≤ d } ]. The Hyperbolic Plane.
After choosing a designated origin O in the two-dimensional hyper-bolic plane, together with a reference ray starting at O , a point p is uniquely identified by its radius r ( p ), denoting the hyperbolic distance to O , and its angle (or angular coordinate ) ϕ ( p ),denoting the angular distance between the reference ray and the line through p and O . Thehyperbolic distance between two points p and q is given bydist( p, q ) = acosh(cosh( r ( p )) cosh( r ( q )) − sinh( r ( p )) sinh( r ( q )) cos(∆ ϕ ( ϕ ( p ) , ϕ ( q )))) , where cosh( x ) = ( e x + e − x ) /
2, sinh( x ) = ( e x − e − x ) / e x / ± o (1)), and∆ ϕ ( p, q ) = π − | π − | ϕ ( p ) − ϕ ( q ) || denotes the angular distance between p and q . If not statedotherwise, we assume that computations on angles are performed modulo 2 π .We use B p ( r ) to denote a disk of radius r centered at p , i.e., the set of points withhyperbolic distance at most r to p . Such a disk has an area of 2 π (cosh( r ) −
1) and circumference2 π sinh( r ). Thus, the area and the circumference of a disk in the hyperbolic plane growexponentially with its radius. In contrast, this growth is polynomial in Euclidean space.Therefore, representing hyperbolic shapes in the Euclidean geometry results in a distortion.In the native representation , used in our figures, circles can appear teardrop-shaped (seeFigure 2). Hyperbolic Random Graphs.
Hyperbolic random graphs are obtained by distributing n points uniformly at random within the disk B O ( R ) and connecting any two of them if andonly if their hyperbolic distance is at most R ; see Figure 1. The disk radius R (which matchesthe connection threshold) is defined as R = 2 log(8 n/ ( π ¯ κ )), where ¯ κ is a constant describingthe desired average degree of the generated network. The coordinates for the vertices aredrawn as follows. For vertex v the angular coordinate, denoted by ϕ ( v ), is drawn uniformlyat random from [0 , π ] and the radius of v , denoted by r ( v ), is sampled according to theprobability density function α sinh( αr ) / (cosh( αR ) −
1) for r ∈ [0 , R ] and α ∈ (1 / , f ( r ) = 12 π α sinh( αr )cosh( αR ) − α π e − α ( R − r ) (1 + Θ( e − αR − e − αr )) , (1)is their joint distribution function for r ∈ [0 , R ]. For r > R , f ( r ) = 0. The constant α ∈ (1 / ,
1) is used to tune the power-law exponent β = 2 α + 1 of the degree distributionof the generated network. Note that we obtain power-law exponents β ∈ (2 , Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs β < β > n . The obtained graphs have logarithmic tree width [4], meaning the VertexCover problem can be solved efficiently in that case.The probability for a given vertex to lie in a certain area A of the disk is given by itsprobability measure µ ( A ) = R A f ( r )d r . The hyperbolic distance between two vertices u and v increases with increasing angular distance between them. The maximum angular distancesuch that they are still connected by an edge is bounded by [14, Lemma 6] θ ( r ( u ) , r ( v )) = arccos (cid:18) cosh( r ( u )) cosh( r ( v )) − cosh( R )sinh( r ( u )) sinh( r ( v )) (cid:19) = 2 e ( R − r ( u ) − r ( v )) / (1 + Θ( e R − r ( u ) − r ( v ) )) . (2) Interval Graphs and Circular Arc Graphs.
In an interval graph each vertex v is identifiedwith an interval on the real line and two vertices are adjacent if and only if their intervalsintersect. The interval width of an interval graph G , denoted by iw( G ), is its maximumclique size, i.e., the maximum number of intervals that intersect in one point. For anygraph the interval width is defined as the minimum interval width over all of its intervalsupergraphs. Circular arc graphs are a superclass of interval graphs, where each vertex isidentified with a subinterval of the circle called circular arc or simply arc . The interval widthof a circular arc graph G is at most twice the size of its maximum clique, since one obtainsan interval supergraph of G by mapping the circular arcs into the interval [0 , π ] on the realline and replacing all intervals that were split by this mapping with the whole interval [0 , π ].Consequently, for any graph G , if k denotes the minimum over the maximum clique numberof all circular arc supergraphs G of G , then the interval width of G is at most 2 k . Treewidth and Pathwidth. A tree decomposition of a graph G is a tree T where each treenode represents a subset of the vertices of G called bag , and the following requirements haveto be satisfied: Each vertex in G is contained in at least one bag, all bags containing agiven vertex in G form a connected subtree of T , and for each edge in G , there exists a bagcontaining both endpoints. The width of a tree decomposition is the size of its largest bagminus one. The treewidth of G is the minimum width over all tree decompositions of G . The path decomposition of a graph is defined analogously to the tree decomposition, with theconstraint that the tree has to be a path. Additionally, as for the treewidth, the pathwidth of a graph G , denoted by pw( G ), is the minimum width over all path decompositions of G .Clearly the pathwidth is an upper bound on the treewidth. It is known that for any graph G and any k ≥
0, the interval width of G is at most k + 1 if and only if its pathwidth is atmost k [8, Theorem 7.14]. Consequently, if k is the maximum clique size of a circular arcsupergraph of G , then 2 k − G . Probabilities.
Since we are analyzing a random graph model, our results are of probabilisticnature. To obtain meaningful statements, we show that they hold with high probability (forshort whp. ), i.e., with probability 1 − O ( n − ). The following Chernoff bound is a useful toolfor showing that certain events occur with high probability. (cid:73) Theorem 1 (Chernoff Bound [11, A.1]) . Let X , . . . , X n be independent random variableswith X i ∈ { , } and let X be their sum. Let f ( n ) = Ω(log( n )) . If f ( n ) is an upper boundfor E [ X ] , then for each constant c there exists a constant c such that X ≤ c f ( n ) holds withprobability − O ( n − c ) . . Bläsius, P. Fischbeck, T. Friedrich, M. Katzmann 5 Figure 1
A hyperbolic random graph with 979 nodes, average degree 8 .
3, and a power-lawexponent of 2 .
5. With high probability, the gray vertices and edges are removed by the dominancereduction rule. Additionally, the remaining subgraph in the outer band (consisting of the blackvertices and edges) has a small path width, with high probability.
Reduction rules are often applied as a preprocessing step, before using a brute force searchor branching in a search tree. They simplify the input by removing parts that are easy tosolve. For example, an isolated vertex does not cover any edges and can thus never be partof a minimum vertex cover. Consequently, in a preprocessing step all isolated vertices can beremoved, which leads to a reduced input size without impeding the search for a minimum.The dominance reduction rule was previously defined for the
IndependentSet prob-lem [12], and later used for
VertexCover in the experiments by Akiba and Iwata [1].Formally, vertex u dominates a neighbor v ∈ N ( u ) if ( N ( v ) \ { u } ) ⊆ N ( u ), i.e., all neighborsof v are also neighbors of u . We say u is dominant if it dominates at least one vertex. Thedominance rule states that u can be added to the vertex cover (and afterwards removedfrom the graph), without impeding the search for a minimum vertex cover. To see that thisis correct, assume that u dominates v and let S be a minimum vertex cover that does notcontain u . Since S has to cover all edges, it contains all neighbors of u . These neighborsinclude v and all of v ’s neighbors, since u dominates v . Therefore, removing v from S leavesonly the edge { u, v } uncovered which can be fixed by adding u instead. The resulting vertexcover has the same size as S . When searching for a minimum vertex cover of G , it is thussafe to assume that u is part of the solution and to reduce the search to G [ V \ { u } ].In the remainder of this section, we study the effectiveness of the dominance reductionrule on hyperbolic random graphs and conclude that VertexCover can be solved efficientlyon these graphs. Our results are summarized in the following main theorem.
Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs u vO B u ( R ) B v ( R ) R u vOR D ( u ) δ ( r ( u ) , r ( v )) Figure 2
Left: Vertex u dominates vertex v , as B v ( R ) ∩ B O ( R ) (light gray) is completely containedin B u ( R ) ∩ B O ( R ) (gray). Right: All vertices that lie in D ( u ) are dominated by u . (cid:73) Theorem 2.
Let G be a hyperbolic random graph on n vertices. Then the VertexCover problem on G can be solved in poly( n ) time, with high probability. The proof of Theorem 2 consists of two parts that make use of the underlying hyperbolicgeometry. In the first part, we show that applying the dominance reduction rule once removesall vertices in the inner part of the hyperbolic disk with high probability, as depicted inFigure 1. We note that this is independent of the order in which the reduction rule is applied,as dominant vertices remain dominant after removing other dominant vertices. In the secondpart, we consider the induced subgraph containing the remaining vertices near the boundaryof the disk (black vertices in Figure 1). We prove that this subgraph has a small pathwidth,by showing that there is a circular arc supergraph with a small interval width. Consequently,a tree decomposition of this subgraph can be computed efficiently. Finally, we obtain apolynomial time algorithm for
VertexCover by first applying the reduction rules andafterwards solving
VertexCover on the remaining subgraph using dynamic programmingon the tree decomposition of small width.
Recall that a hyperbolic random graph is obtained by distributing n vertices in a hyperbolicdisk B O ( R ) and that any two are connected if their distance is at most R . Consequently,one can imagine the neighborhood of a vertex u as another disk B u ( R ). Vertex u dominatesanother vertex v if its neighborhood disk completely contains that of v (both constrainedto B O ( R )), as depicted in Figure 2 left. We define the dominance area D ( u ) of u to bethe area containing all such vertices v . That is, D ( u ) = { p ∈ B O ( R ) | B p ( R ) ∩ B O ( R ) ⊆ B u ( R ) ∩ B O ( R ) } . The result is illustrated in Figure 2 right. We note that it is sufficient fora vertex v to lie in D ( u ) in order to be dominated by u , however, it is not necessary.Given the radius r ( u ) of vertex u we can now compute a lower bound on the probabilitythat u dominates another vertex, i.e., the probability that at least one vertex lies in D ( u ),by determining the measure µ ( D ( u )). To this end, we first define δ ( r ( u ) , r ( v )) to be themaximum angular distance between two nodes u and v such that v lies in D ( u ). (cid:73) Lemma 3.
Let u, v be vertices with r ( u ) ≤ r ( v ) . Then, v ∈ D ( u ) if ∆ ϕ ( u, v ) is at most δ ( r ( u ) , r ( v )) = 2( e − r ( u ) / − e − r ( v ) / ) + Θ( e − / r ( u ) ) − Θ( e − / r ( v ) ) . Proof.
Without loss of generality we assume that ϕ ( u ) = 0. For now assume that ϕ ( v ) = ϕ ( u ).Since r ( v ) ≥ r ( u ) we know that the intersections of the boundaries of B v ( R ) with B O ( R ) lie . Bläsius, P. Fischbeck, T. Friedrich, M. Katzmann 7 O u v i u i v R δ ( r ( u ) , r ( v )) B u ( R ) B v ( R ) Figure 3
Vertex u dominates vertex v , with r ( u ) ≤ r ( v ), if ∆ ϕ ( u, v ) ≤ ∆ ϕ ( i u , i v ). between those of B u ( R ) with B O ( R ), as is depicted in Figure 3. Now let i u denote one ofthese intersections for B u ( R ) and B O ( R ), and let i v denote the intersection for B v ( R ) and B O ( R ) that is on the same side of the ray through O and u as i u . It is easy to see that themaximum angular distance between u and v such that B v ( R ) ∩ B O ( R ) is contained within B u ( R ) ∩ B O ( R ) is given by the angular distance between i u and i v . Therefore, v lies in thedominance area of u if ∆ ϕ ( u, v ) ≤ ∆ ϕ ( i u , i v ).Recall that θ ( r ( p ) , r ( q )) denotes the maximum angular distance such that dist( p, q ) ≤ R ,as defined in Equation (2). Since i u and i v have radius R and hyperbolic distance R to u and v , respectively, we know that their angular coordinates are θ ( r ( u ) , R ) and θ ( r ( v ) , R ),respectively. Consequently, the angular distance between i u and i v is given by δ ( r ( u ) , r ( v )) = θ ( r ( u ) , R ) − θ ( r ( v ) , R )= 2( e − r ( u ) / − e − r ( v ) / ) + Θ( e − / r ( u ) ) − Θ( e − / r ( v ) ) . (cid:74) Using Lemma 3 we can now compute the probability for a given vertex to lie in thedominance area of u . We note that this probability grows roughly like 2 /πe − r ( u ) / , which isa constant fraction of the measure of the neighborhood disk of u which grows as α/ ( α − / · /πe − r ( u ) / [14, Lemma 3.2]. Consequently, the expected number of nodes that u dominatesis a constant fraction of the expected number of its neighbors. (cid:73) Lemma 4.
Let u be a node with radius r ( u ) ≥ R/ . The probability for a given node tolie in D ( u ) is given by µ ( D ( u )) = 2 π e − r ( u ) / (1 − Θ( e − α ( R − r ( u )) )) ± O (1 /n ) . Proof.
The probability for a given vertex v to lie in D ( u ) is obtained by integrating theprobability density (given by Equation (1)) over D ( u ). µ ( D ( u )) = 2 Z Rr ( u ) Z δ ( r ( u ) ,r )0 f ( r ) d ϕ d r = 2 Z Rr ( u ) (cid:16) e − r ( u ) / − e − r/ ) + Θ( e − / r ( u ) ) − Θ( e − / r ) (cid:17) · α π e − α ( R − r ) (1 + Θ( e − αR − e − αr )) d r Since r ( u ) ≥ R/ r ∈ [ r ( u ) , R ] we have Θ( e − / r ( u ) ) − Θ( e − / r ) = ±O ( e − / R ) and(1 + Θ( e − αR − e − αr )) = (1 + Θ( e − αR )). Due to the linearity of integration, constant factors Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs within the integrand can be moved out of the integral, which yields µ ( D ( u )) = απ e − αR (1 + Θ( e − αR )) Z Rr ( u ) (cid:16) e − r ( u ) / − e − r/ ) ± O ( e − / R ) (cid:17) · e αr d r = 2 απ e − r ( u ) / e − αR (1 + Θ( e − αR )) Z Rr ( u ) e αr d r − απ e − αR (1 + Θ( e − αR )) Z Rr ( u ) e ( α − / r d r ± O e − (3 / α ) R Z Rr ( u ) e αr d r ! . The remaining integrals can be computed easily and we obtain µ ( D ( u )) = 2 π e − r ( u ) / (1 + Θ( e − αR ))(1 − e − α ( R − r ( u )) ) − α ( α − / π e − R/ (1 + Θ( e − αR ))(1 − e − ( α − / R − r ( u )) ) ± O (cid:16) e − / R (1 − e − α ( R − r ( u )) ) (cid:17) . As e − R/ = Θ( n − ) and e − / R = Θ( n − / ), simplifying the error terms yields the claim. (cid:74) The following lemma shows that, with high probability, all vertices that are not too closeto the boundary of the disk dominate at least one vertex. (cid:73)
Lemma 5.
Let G be a hyperbolic random graph with average degree ¯ κ . Then there is aconstant c > / ¯ κ , such that all vertices u with r ( u ) ≤ ρ = R − n c ) are dominant,with high probability. Proof.
Vertex u is dominant if at least one vertex lies in D ( u ). To show this for any u with r ( u ) ≤ ρ , it suffices to show it for r ( u ) = ρ , since D ( u ) increases with decreasing radius. Todetermine the probability that at least one vertex lies in D ( u ), we use Lemma 4 and obtain µ ( D ( u )) = 2 π e − ρ/ (1 − Θ( e − α ( R − ρ ) )) ± O (1 /n )= 2 π e − R/ n c ) (1 − Θ( e − α log log( n c ) )) ± O (1 /n ) . By substituting R = 2 log(8 n/ ( π ¯ κ )), we obtain µ ( D ( u )) = ¯ κ/ (4 n )( c log( n )(1 − o (1)) ± O (1)).The probability of at least one node falling into D ( u ) is now given byPr[ { v ∈ D ( u ) } 6 = ∅ ] = 1 − (1 − µ ( D ( u ))) n ≥ − e − nµ ( D ( u )) = 1 − Θ( n − c ¯ κ/ − o (1)) ) . Consequently, for large enough n we can choose c > / ¯ κ such that the probability of a vertexat radius ρ being dominant is at least 1 − Θ( n − ), allowing us to apply union bound. (cid:74)(cid:73) Corollary 6.
Let G be a hyperbolic random graph and c > / ¯ κ . With high probability, allvertices with radius at most ρ = R − n c ) are removed by the dominance rule. By Corollary 6 the dominance rule removes all vertices of radius at most ρ . Consequently,all remaining vertices have radius at least ρ . We refer to this part of the disk as outer band .More precisely, the outer band is defined as B O ( R ) \ B O ( ρ ). It remains to show that thepathwidth of the subgraph induced by the vertices in the outer band is small. . Bläsius, P. Fischbeck, T. Friedrich, M. Katzmann 9 O r vI v R/ R A R/ A r Figure 4
Left: The circular arcs representing the neighborhood of a vertex. For vertex v the areacontaining the whole neighborhood of v , as well as the circular arc I v are drawn in the same color.Right: The area that contains the vertices whose arcs intersect angle 0. Area A r contains all suchvertices with radius at least r . Vertex v lies on the boundary of A r and its interval I v extends to 0. In the following, we use G r = G [ { v ∈ V } | r ( v ) ≥ r ] to denote the induced subgraph of G thatcontains all vertices with radius at least r . To show that the pathwidth of G ρ (the inducedsubgraph in the outer band) is small, we first show that there is a circular arc supergraph G Sρ of G ρ with a small maximum clique. We use G S to denote a circular arc supergraph of ahyperbolic random graph G , which is obtained by assigning each vertex v an angular interval I v on the circle, such that the intervals of two adjacent vertices intersect. More precisely,for a vertex v , we set I v = [ ϕ ( v ) − θ ( r ( v ) , r ( v )) , ϕ ( v ) + θ ( r ( v ) , r ( v ))]. Intuitively, this meansthat the interval of a vertex contains a superset of all its neighbors that have a larger radius,as can be seen in Figure 4 left. The following lemma shows that G S is actually a supergraphof G . (cid:73) Lemma 7.
Let G = ( V, E ) be a hyperbolic random graph. Then G S is a supergraph of G . Proof.
Let { u, v } ∈ E be any edge in G . To show that G S is a supergraph of G we needto show that u and v are also adjacent in G S , i.e., I u ∩ I v = ∅ . Without loss of generalityassume r ( u ) ≤ r ( v ). Since u and v are adjacent in G , the hyperbolic distance between themis at most R . It follows, that their angular distance ∆ ϕ ( u, v ) is bounded by θ ( r ( u ) , r ( v )).Since θ ( r ( u ) , r ( v )) ≤ θ ( r ( u ) , r ( u )) for r ( u ) ≤ r ( v ), we have ∆ ϕ ( u, v ) ≤ θ ( r ( u ) , r ( u )). As I u extends by θ ( r ( u ) , r ( u )) from ϕ ( u ) in both directions, it follows that ϕ ( v ) ∈ I u . (cid:74) It is easy to see that, after removing a vertex from G and G S , G S is still a supergraphof G . Consequently, G Sρ is a supergraph of G ρ . It remains to show that G Sρ has a smallmaximum clique number, which is given by the maximum number of arcs that intersect atany angle. To this end, we first compute the number of arcs that intersect a given anglewhich we set to 0 without loss of generality. Let A r denote the area of the disk containing allvertices v with radius r ( v ) ≥ r whose interval I v intersects 0, as illustrated in Figure 4 right.The following lemma describes the probability for a given vertex to lie in A r . (cid:73) Lemma 8.
Let G be a hyperbolic random graph and let r ≥ R/ . The probability for agiven vertex to lie in A r is bounded by µ ( A r ) ≤ α (1 − α ) π e − ( α − / R − (1 − α ) r · (cid:16) e − αR + e − (2 r − R ) − e − (1 − α )( R − r ) ) (cid:17) . Proof.
We obtain the measure of A r by integrating the probability density function over A r .Due to the definition of I v we can conclude that A r includes all vertices v with radius r ( v ) ≥ r whose angular distance to 0 is at most θ ( r ( v ) , r ( v )), defined in Equation (2). We obtain, µ ( A r ) = Z Rr Z θ ( x,x )0 f ( x ) d ϕ d x = 2 Z Rr e ( R − x ) / (1 ± Θ( e R − x )) · α π e − α ( R − x ) (1 + Θ( e − αR − e − αx )) d x. As before, we can conclude that (1 + Θ( e − αR − e − αr )) = (1 + Θ( e − αR )), since r ≥ R/
2. Bymoving constant factors out of the integral, the expression can be simplified to µ ( A r ) ≤ απ e − ( α − / R (1 + Θ( e − αR )) Z Rr e − (1 − α ) x (1 + Θ( e R − x )) d x. We split the sum in the integral and deal with the two resulting integrals separately. µ ( A r ) ≤ απ e − ( α − / R (1 + Θ( e − αR )) Z Rr e − (1 − α ) x d x + Θ Z Rr e − (1 − α ) x + R − x d x !! = 2 απ e − ( α − / R (1 + Θ( e − αR )) · − α e − (1 − α ) r (1 − e − (1 − α )( R − r ) ) + Θ (cid:16) e R e − (3 − α ) r (1 − e − (3 − α )( R − r ) ) (cid:17) ! . By placing 1 / (1 − α ) e − (1 − α ) r outside of the brackets we obtain µ ( A r ) ≤ α (1 − α ) π e − ( α − / R − (1 − α ) r (1 + Θ( e − αR )) · (1 − e − (1 − α )( R − r ) ) + Θ (cid:16) e R − r (1 − e − (3 − α )( R − r ) ) (cid:17) ! . Simplifying the remaining error terms then yields the claim. (cid:74)
We can now bound the maximum clique number in G Sρ and thus its interval width iw( G Sρ ). (cid:73) Theorem 9.
Let G be a hyperbolic random graph and r ≥ R/ . Then there exists aconstant c such that, whp., iw( G Sr ) = O (log( n )) if r ≥ R − − α ) log log( n c ) , and otherwise iw( G Sr ) ≤ α (1 − α ) π ne − ( α − / R − (1 − α ) r (cid:16) e − αR + e − (2 r − R ) − e − (1 − α )( R − r ) ) (cid:17) . Proof.
We start by determining the expected number of arcs that intersect at a given angle,which can be done by computing the expected number of vertices in A r , using Lemma 8: E [ |{ v ∈ A r }| ] ≤ α (1 − α ) π ne − ( α − / R − (1 − α ) r (1 + Θ( e − αR + e − (2 r − R ) − e − (1 − α )( R − r ) )) . It remains to show that this bound holds with high probability at every angle. To thisend, we make use of a Chernoff bound (Theorem 1), by first showing that the bound on . Bläsius, P. Fischbeck, T. Friedrich, M. Katzmann 11 E [ |{ v ∈ A r }| ] is Ω(log( n )). We start with the case where r < R − − α log log( n c ). E [ |{ v ∈ A r }| ] < α (1 − α ) π ne − ( α − / R − (1 − α )( R − / (1 − α ) log log( n c )) · (cid:16) e − αR + e − (2( R − / (1 − α ) log log( n c )) − R ) − e − (1 − α )( R − ( R − / (1 − α ) log log( n c ))) ) (cid:17) = 2 α (1 − α ) π ne − R/ n c )) · (cid:16) e − αR + e − ( R − / (1 − α ) log log( n c )) − e − log log( n c ) ) (cid:17) Substituting R = 2 log(8 n/ ( π ¯ κ )) we obtain E [ |{ v ∈ A r }| ] < α ¯ κc − α ) log( n )(1 + o (1)) . Thus, for all radii smaller than R − − α ) log log( n c ), the resulting upper bound is lowerbounded by Ω(log( n )), which lets us apply Theorem 1. Moreover, as E [ |{ v ∈ A r }| ] decreaseswith increasing r , O (log( n )) is a pessimistic but valid upper bound for the case r ≥ R − − α ) log log( n c ). Thus, we can also apply Theorem 1 to this case, using the O (log( n )) bound.By Theorem 1, we can choose c such that in both cases the bound holds with probability1 − O ( n − c ) for any c at a given angle. In order to see that it holds at every angle, note thatit suffices to show that it holds at all arc endings as the number of intersecting arcs does notchange in between arc endings. Since there are exactly 2 n arc endings, we can apply unionbound and obtain that the bound holds with probability 1 − O ( n − c +1 ) for any c at everyangle. Since our bound on E [ |{ v ∈ A r }| ] is an upper bound on the maximum clique size of G Sr , the interval width of G Sr is at most twice as large, as argued in Section 2. (cid:74) Since the interval width of a circular arc supergraph of G is an upper bound on thepathwidth of G [8, Theorem 7.14] and since ρ ≥ R − / (1 − α ) log log( n c ) for α ∈ (1 / , (cid:73) Corollary 10.
Let G be a hyperbolic random graph and let G ρ be the subgraph obtained byremoving all vertices with radius at most ρ = R − n c ) . Then, pw( G ρ ) = O (log( n )) . We are now ready to prove our main theorem, which we restate for the sake of readability. (cid:73)
Theorem 2.
Let G be a hyperbolic random graph on n vertices. Then the VertexCover problem in G can be solved in poly( n ) time, with high probability. Proof.
Consider the following algorithm that finds the minimum vertex cover of G . Westart with an empty vertex cover S . Initially, all dominant vertices are added to S , whichis correct due to the dominance rule. By Lemma 5, this includes all vertices of radius atmost ρ = R − n c ), for some constant c , with high probability. Obviously, finding allvertices that are dominant can be done in poly( n ) time. It remains to determine a vertexcover of G ρ . By Corollary 10, the pathwidth of G ρ is O (log( n )), with high probability. Sincethe pathwidth is an upper bound on the treewidth, we can find a tree decomposition of G ρ and solve the VertexCover problem in G ρ in poly( n ) time [8, Theorems 7.18 and 7.9]. (cid:74) Moreover, linking the radius of a vertex in Theorem 9 with its expected degree leadsto the following corollary, which is interesting in its own right. It links the pathwidth tothe degree d in the graph G ≤ d . Recall that G ≤ d denotes the subgraph of G induced by thevertices of degree at most d . (cid:73) Corollary 11.
Let G be a hyperbolic random graph and let d ≤ √ n . Then, with highprobability, pw( G ≤ d ) = O ( d − α + log( n )) . Proof.
Consider the radius r = R − εd ) for some constant ε >
0, and the graph G r whichis obtained by removing all vertices of radius at most r . By substituting R = 2 log(8 n/ ( π ¯ κ ))and using [14, Lemma 3.2] we can compute the expected degree of a vertex with radius r as E [deg( v ) | r ( v ) = r ] = 2 α ( α − / π ne − r/ (1 ± O ( e − ( α − / r )) = α ¯ κε α − / d (1 ± o (1)) . First assume that d ≥ log( n ) / (2 − α ) . We handle the other case later. Since d ∈ Ω(log( n ))we can choose ε large enough to apply Theorem 1 and conclude that this holds with highprobability. Furthermore, since a smaller radius implies a larger degree, we know that, withhigh probability, all nodes v with radius at most r , havedeg( v ) ≥ α ¯ κε α − / d (1 ± o (1)) . For large enough n we can choose ε such that, with high probability, G r is a supergraph of G ≤ d .To prove the claim, it remains to bound the pathwidth of G r . If r > R − / (1 − α ) log log( n c ),we can apply the first part of Theorem 9 to obtain iw( G Sr ) = O (log( n )). Otherwise, we usepart two to conclude that the interval width of G r is at mostiw( G Sr ) ≤ α (1 − α ) π ne − ( α − / R − (1 − α ) r (cid:16) e − αR + e − (2 r − R ) − e − (1 − α )( R − r ) ) (cid:17) = α ¯ κε − α (2 − α ) d − α (cid:16) n − α + (( εd ) /n ) − ( εd ) − (2 − α ) ) (cid:17) = O ( d − α ) . As argued in Section 2 the interval width of a graph is an upper bound on the pathwidth.For d < log( n ) / (2 − α ) (which we excluded above), consider G ≤ d for d = log( n ) / (2 − α ) >d . As we already proved the corollary for d , we obtain pw( G ≤ d ) = O ( d − α + log( n )) = O (log( n )). As G ≤ d is a subgraph of G ≤ d , the same bound holds for G ≤ d . (cid:74) Our results show that a heterogeneous degree distribution as well as high clustering makethe dominance rule very effective. This matches the behavior for real-world networks, whichtypically exhibit these two properties. However, our analysis actually makes more specificpredictions: (I) vertices with sufficiently high degree usually have at least one neighbor theydominate and can thus safely be included in the vertex cover; and (II) the graph remainingafter deleting the high degree vertices has simple structure, i.e., small pathwidth.To see whether this matches the real world, we run experiments on 59 networks fromseveral network datasets [2, 3, 18, 19, 20]. Although the focus of this paper is the theoreticalanalysis on hyperbolic random graphs, we briefly report on our experimental results; seeTable 1 in Appendix 5. Out of the 59 instances, we can solve
VertexCover for 47 networksin reasonable time. We refer to these as easy , while the remaining 12 are called hard . Notethat our theoretical analysis aims at explaining why the easy instances are easy.Recall from Lemma 5 that all vertices with radius at most R − n / ¯ κ ) probablydominate, which corresponds to an expected degree of α/ ( α − / · log n . For more than halfof the 59 networks, more than 78 % of the vertices above this degree were in fact dominant.For more than a quarter of the networks, more than 96 % were dominant. Restricted to the47 easy instances, these number increase to 82 % and 99 %, respectively. . Bläsius, P. Fischbeck, T. Friedrich, M. Katzmann 13 Experiments concerning the pathwidth of the resulting graph are much more difficult, dueto the lack of efficient tools. Therefore, we used the tool by Tamaki et al. [21] to heuristicallycompute upper bounds on the treewidth instead. As in our analysis, we only removed verticesthat dominate in the original graph instead of applying the reduction rule exhaustively. Onthe resulting subgraphs, the treewidth heuristic ran with a 15 min timeout. The resultingtreewidth is at most 50 for 44 % of the networks, at most 15 for 34 %, and at most 5 for 25 %.Restricted to easy instances, the values increase to 55 %, 43 %, and 32 %, respectively.Hyperbolic random graphs are of course an idealized representation of real-world networks.However, these experiments indicate that the predictions derived from the model match thereal world, at least for a significant fraction of networks.
Approximation.
Concerning approximation algorithms for
VertexCover , there is a similartheory-practice gap as for exact solutions. In theory, there is a simple 2-approximation andthe best known polynomial time approximation reduces the factor to 2 − Θ(log( n ) − / ) [15].However, it is NP-hard to approximate VertexCover within a factor of 1 . − ε for all ε > n approximation. However, on scale-free networksthis strategy performs exceptionally well with approximation ratios very close to 1 [9].Our results for hyperbolic random graphs at least partially explain this good approximationratio. Lemma 5 states that, with high probability, we do not make any mistake by taking allvertices below a certain radius ρ , which corresponds to vertices of at least logarithmic degree.The same computation for larger values of ρ does no longer give such strong guarantees.However, it still gives bounds on the probability for making a mistake. In fact, this errorprobability is sub-constant as long as the corresponding expected degree is super-constant.Although this is not a formal argument, it still explains to a degree why greedy works sowell on networks with a heterogeneous degree distribution and high clustering. Moreover, itindicates how the greedy algorithm should be adapted to obtain better approximation ratios:As the probability to make a mistake grows with growing radius and thus with shrinkingvertex degree, the majority of mistakes are done when all vertices have already low degree.However, for hyperbolic random graphs, the subgraphs induced by vertices below a certainconstant degree decompose into small components for n → ∞ . It thus seems to be a goodidea to run the greedy algorithm only until all remaining vertices have low degree, say k . Theremaining small connected components of maximum-degree k can then be solved with bruteforce in reasonable time. In the following we call the resulting algorithm k -adaptive greedy .We ran experiments on the 47 easy real networks mentioned above (for the hard instances,we cannot measure approximation ratios). For these networks, we compare the normalgreedy algorithm with 2- and 4-adaptive greedy. Note that 2-adaptive greedy is special, as VertexCover can be solved efficiently on graphs with maximum degree 2 (no brute-forcingis necessary). For 4-adaptive greedy, the size of the largest connected component is relevant.The median approximation ratio for greedy over all 47 networks is 1 . .
005 for 2-adaptive and to 1 .
002 for 4-adaptive greedy. Thus, the number of too manyselected vertices goes down by a factor of 1 . . . References Takuya Akiba and Yoichi Iwata. Branch-and-reduce exponential/FPT algorithms in practice:A case study of vertex cover.
Theor. Comput. Sci. , 609:211 – 225, 2016. doi:10.1016/j.tcs.2015.09.023 . Alexandre Arenas, Albert-László Barabási, Vladimir Batagelj, Andrej Mrvar, Mark Newman,and Tore Opsahl. Gephi datasets. https://github.com/gephi/gephi/wiki/Datasets . Vladimir Batagelj and Andrej Mrvar. Pajek datasets. http://vlado.fmf.uni-lj.si/pub/networks/data/ , 2006. Thomas Bläsius, Tobias Friedrich, and Anton Krohmer. Hyperbolic Random Graphs: Separ-ators and Treewidth. In , pages15:1 – 15:16, 2016. doi:10.4230/LIPIcs.ESA.2016.15 . Marián Boguná, Fragkiskos Papadopoulos, and Dmitri Krioukov. Sustaining the internet withhyperbolic mapping.
Nat. Commun. , 1:62, 2010. doi:10.1038/ncomms1063 . Liming Cai and David Juedes. On the existence of subexponential parameterized algorithms.
J. Comput. Syst. Sci. , 67:789 – 807, 2003. doi:10.1016/S0022-0000(03)00074-6 . Jianer Chen, Iyad A. Kanj, and Ge Xia. Improved upper bounds for vertex cover.
Theor.Comput. Sci. , 411(40):3736 – 3756, 2010. doi:10.1016/j.tcs.2010.06.026 . Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, MarcinPilipczuk, Michał Pilipczuk, and Saket Saurabh.
Parameterized Algorithms . Springer, 2015. Mariana O. Da Silva, Gustavo A. Gimenez-Lugo, and Murilo V. G. Da Silva. Vertexcover in complex networks.
Int. J. Mod. Phys. C , 24(11):1350078, 2013. doi:10.1142/S0129183113500782 . Irit Dinur and Samuel Safra. On the hardness of approximating minimum vertex cover.
Ann.Math. , 162(1):439 – 485, 2005. doi:10.4007/annals.2005.162.439 . Devdatt P. Dubhashi and Alessandro Panconesi.
Concentration of Measure for the Analysisof Randomized Algorithms . Cambridge University Press, 2012. Fedor V. Fomin, Fabrizio Grandoni, and Dieter Kratsch. A measure & conquer approach for theanalysis of exact algorithms.
J. ACM , 56(5):25:1 – 25:32, 2009. doi:10.1145/1552285.1552286 . Tobias Friedrich and Anton Krohmer. On the diameter of hyperbolic random graphs. In
Automata, Languages, and Programming , pages 614 – 625. Springer Berlin Heidelberg, 2015. doi:10.1007/978-3-662-47666-6_49 . Luca Gugelmann, Konstantinos Panagiotou, and Ueli Peter. Random hyperbolic graphs:Degree sequence and clustering. In
Automata, Languages, and Programming , pages 573 – 585.Springer Berlin Heidelberg, 2012. doi:10.1007/978-3-642-31585-5_51 . George Karakostas. A better approximation ratio for the vertex cover problem.
ACM Trans.Algorithms , 5(4):41:1 – 41:8, 2009. doi:10.1145/1597036.1597045 . Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2 − ε . J. Comput. Syst. Sci. , 74(3):335 – 349, 2008. doi:10.1016/j.jcss.2007.06.019 . Dmitri Krioukov, Fragkiskos Papadopoulos, Maksim Kitsak, Amin Vahdat, and MariánBoguñá. Hyperbolic geometry of complex networks.
Phys. Rev. E , 82:036106, 2010. doi:10.1103/PhysRevE.82.036106 . Jérôme Kunegis. KONECT: The koblenz network collection. In
International Conference onWorld Wide Web (WWW) , pages 1343 – 1350, 2013. doi:10.1145/2487788.2488173 . Jure Leskovec and Andrej Krevl. SNAP Datasets: Stanford large network dataset collection. http://snap.stanford.edu/data , 2014. Ryan A. Rossi and Nesreen K. Ahmed. The network data repository with interactive graphanalytics and visualization. In
Proceedings of the Twenty-Ninth AAAI Conference on ArtificialIntelligence , 2015. URL: http://networkrepository.com . Hisao Tamaki, Hiromu Ohtsuka, Takuto Sato, and Keitaro Makii. TCS-Meiji PACE2017-TrackA. github.com/TCS-Meiji/PACE2017-TrackA , 2017. Mingyu Xiao and Hiroshi Nagamochi. Exact algorithms for maximum independent set.
Inf.Comput. , 255:126 – 146, 2017. doi:10.1016/j.ic.2017.06.001 . . Bläsius, P. Fischbeck, T. Friedrich, M. Katzmann 15 Table 1 (continuing on the next page) shows the raw data of our experiments for which wereported aggregate values in the discussion in Section 4. The percentage of dominant verticesamong those with high degree (over α/ ( α − / · log n ) is rounded to whole percentages.The approximation ratios are rounded to three decimal digits. Treewidth − Table 1
The raw data of our experiments. The columns are: (network) the network’s name; (easy) whether or not we could compute an optimal solution; (dom) the percentage of dominantvertices among the high-degree vertices; (tw) an upper bound for the treewidth of the remaininggraph after deleting dominant nodes; (greedy) the approximation ratio of greedy; (2-ad) theapproximation ratio of 2-adaptive greedy; (4-ad) the approximation ratio of 4-adaptive greedy; (comp) the size of the largest component that remains after the greedy phase of 4-adaptive greedy. network easy dom tw greedy 2-ad 4-ad comp advogato (cid:51)
51 % 314 1.011 1.009 1.005 863airlines (cid:51)
28 % 23 1.000 1.000 1.000 75as-22july06 (cid:51)
100 % 3 1.002 1.001 1.001 46as-caida20071105 (cid:51)
100 % 3 1.002 1.001 1.000 35as-skitter (cid:55)
47 % 969794as20000102 (cid:51)
100 % 2 1.003 1.001 1.001 18bio-CE-HT (cid:51)
100 % 3 1.015 1.009 1.000 225bio-CE-LC (cid:51)
100 % 2 1.003 1.003 1.003 39bio-DM-HT (cid:51)
50 % 13 1.017 1.014 1.004 319bio-yeast-protein-inter (cid:51)
100 % 4 1.013 1.006 1.002 147bn-fly-drosophila-medulla-1 (cid:51)
72 % 38 1.018 1.013 1.009 142bn-mouse-kasthuri-graph-v4 (cid:51)
100 % 1 1.006 1.000 1.000 12ca-AstroPh (cid:51)
94 % 6 1.003 1.002 1.001 123ca-cit-HepPh (cid:51)
84 % 151 1.003 1.003 1.002 533ca-CondMat (cid:51)
99 % 4 1.003 1.002 1.001 53ca-GrQc (cid:51)
99 % 2 1.004 1.002 1.001 44ca-HepTh (cid:51)
95 % 13 1.005 1.004 1.001 174cfinder-google (cid:55)
66 % 82cit-HepTh (cid:55)
13 % 19737citeseer (cid:55)
46 % 182372com-amazon (cid:51)
93 % 2756 1.011 1.006 1.002 16209com-dblp (cid:51)
100 % 7 1.002 1.001 1.000 69cpan-authors (cid:51)
100 % 2 1.009 1.009 1.009 17digg-friends (cid:51)
58 % 1649 1.008 1.006 1.004 179ego-facebook (cid:51)
100 % -1 1.000 1.000 1.000 3ego-gplus (cid:51)
100 % 1 1.000 1.000 1.000 5email-Enron (cid:51)
85 % 41 1.003 1.002 1.001 141EuroSiS (cid:51)
56 % 34 1.020 1.018 1.010 274facebook-wosn-links (cid:55)
27 % 36694flixster (cid:55)
73 % 122hyves (cid:51)
98 % 1653 1.008 1.008 1.008 42livemocha (cid:51) (cid:51)
76 % 619 1.014 1.009 1.004 4658
Table 1
The raw data of our experiments. The columns are: (network) the network’s name; (easy) whether or not we could compute an optimal solution; (dom) the percentage of dominantvertices among the high-degree vertices; (tw) an upper bound for the treewidth of the remaininggraph after deleting dominant nodes; (greedy) the approximation ratio of greedy; (2-ad) theapproximation ratio of 2-adaptive greedy; (4-ad) the approximation ratio of 4-adaptive greedy; (comp) the size of the largest component that remains after the greedy phase of 4-adaptive greedy. network easy dom tw greedy 2-ad 4-ad comp loc-gowalla-edges (cid:55)
64 % 3991moreno-names (cid:51)
94 % 3 1.006 1.004 1.002 34moreno-propro (cid:51)
100 % 4 1.014 1.006 1.002 153munmun-twitter-social (cid:51)
57 % 12 1.000 1.000 1.000 5OClinks (cid:51)
36 % 202 1.017 1.015 1.005 498p2p-Gnutella04 (cid:51)
42 % 1352 1.019 1.017 1.016 970p2p-Gnutella05 (cid:51)
40 % 1075 1.014 1.013 1.013 447p2p-Gnutella06 (cid:51)
40 % 1142 1.023 1.022 1.021 820p2p-Gnutella08 (cid:51)
47 % 414 1.008 1.008 1.008 45p2p-Gnutella09 (cid:51)
47 % 419 1.005 1.005 1.005 63p2p-Gnutella24 (cid:51)
81 % 525 1.006 1.005 1.005 70p2p-Gnutella25 (cid:51)
79 % 464 1.006 1.005 1.005 77p2p-Gnutella30 (cid:51)
79 % 604 1.005 1.005 1.004 62p2p-Gnutella31 (cid:51)
80 % 732 1.011 1.010 1.010 65petster-carnivore (cid:51)
79 % 149312 1.008 1.007 1.004 9238petster-friendship-cat (cid:55)
12 % 14929petster-friendship-dog (cid:55)
15 % 340634petster-friendship-hamster (cid:55)
23 % 135soc-Epinions1 (cid:51)
82 % 238 1.006 1.003 1.001 228US-Air (cid:51)
67 % 4 1.013 1.000 1.000 23web-Google (cid:55)
84 % 103939wiki-Vote (cid:51)
44 % 384 1.054 1.052 1.050 726wordnet-words (cid:51)
95 % 28 1.004 1.003 1.002 59YeastS (cid:51)
70 % 39 1.013 1.012 1.005 244youtube-links (cid:51)
86 % 1239 1.008 1.004 1.001 570youtube-u-growth (cid:55)(cid:55)