Hyperbolic intersection graphs and (quasi)-polynomial time
We study unit ball graphs (and, more generally, so-called noisy uniform ball graphs) in -dimensional hyperbolic space, which we denote by . Using a new separator theorem, we show that unit ball graphs in enjoy similar properties as their Euclidean counterparts, but in one dimension lower: many standard graph problems, such as Independent Set, Dominating Set, Steiner Tree, and Hamiltonian Cycle can be solved in time for any fixed , while the same problems need time in . We also show that these algorithms in are optimal up to constant factors in the exponent under ETH.
This drop in dimension has the largest impact in , where we introduce a new technique to bound the treewidth of noisy uniform disk graphs. The bounds yield quasi-polynomial () algorithms for all of the studied problems, while in the case of Hamiltonian Cycle and -Coloring we even get polynomial time algorithms. Furthermore, if the underlying noisy disks in have constant ply, then all studied problems can be solved in polynomial time. This contrasts with the fact that these problems require time under ETH in constant-ply Euclidean unit disk graphs.
Finally, we complement our quasi-polynomial algorithm for Independent Set in noisy uniform disk graphs with a matching lower bound under ETH. As far as we know, this is the first natural problem with a quasi-polynomial lower bound that is shown to be tight.
Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlandss.firstname.lastname@example.org \CopyrightSándor Kisfaludi-Bak\subjclass \ccsdescTheory of computation Computational geometry \relatedversion\fundingThis work was supported by the Netherlands Organization for Scientific Research NWO under project no. 024.002.003.\EventShortTitleArXiv \EventAcronymArXiv
Hyperbolic space has seen increasing interest in recent years from various communities in computer science due to its unique metric properties; examples include the study of random networks [19, 3, 21], routing and load balancing [25, 35], metric embeddings [34, 38], and visualization . The algorithmic properties of hyperbolic space have been mostly studied through Gromov’s hyperbolicty, which is a convenient combinatorial description for negatively curved spaces [26, 10, 18].
Treewidth is one of the most celebrated graph parameters, because it seems to capture the class of graphs where dynamic-programming-based solutions work efficiently. It is also closely related to separators and divide-and-conquer algorithms.
In this paper, we study treewidth bounds for intersection graphs of unit-ball-like objects in hyperbolic space; these intersection graphs are a natural choice to capture some important properties of the underlying metric. The most well-studied geometric intersection graphs are unit disk graphs in . From the perspective of treewidth and exact algorithms, unit disk graphs have some intriguing properties: they are potentially dense, (may have treewidth ), but they still exhibit the “square root phenomenon” for several problems just as planar and minor-free graphs do; so for example one can solve Independent Set or -coloring in these classes in time , while these problems would require time in general graphs unless the Exponential Time Hypothesis (ETH)  fails. In , the best Independent Set running time for unit ball graphs is [29, 13].
To our knowledge, the only paper studying treewidth of graphs in hyperbolic space is the work by Bläsius, Friedrich, and Krohmer , where they investigate random hyperbolic uniform disk graphs chosen from some distributions, and prove various treewidth bounds depending on the parameter of the distribution. Our goal is to get worst case bounds on the treewidth of intersection graphs. Naturally, getting a sublinear bound on separators or treewidth itself is not possible, since cliques are unit ball graphs. Therefore, we use the partition and weighting scheme developed by De Berg et al. . Given an intersection graph , one defines a partition of the vertex set ; initially, it is useful to think of a partition into cliques using a tiling of the underlying space where each tile has a small diameter. Then the partition classes are defined for each non-empty tile as the set of balls whose center falls inside the tile. This gives a clique partition of . Next, each clique receives a weight of . It turns out that this weighting is useful for many problems, and it motivates us to define weighted separators of with respect to as a separator consisting of classes of , whose weight is defined as the sum of the weights of the constituent partition classes. It is in this sense that we can find sublinear separators.
Such weightings can also be used along with treewidth techniques. Let be a graph and let be a partition of . Let be the graph obtained by contracting all edges of that go within a partition class of , i.e., can be identified with . For each class , we assign the weight . We define the -flattened treewidth of as the weighted treewidth of under this weighting. (The definitions of treewidth, weighted treewidth, and various partition schemes are recalled in Section 3.1.) For example, unit ball graphs in have a clique-partition such that their -flattened treewidth is .
We intend to show that unit disk graphs in hyperbolic space are even more intriguing than the ones in Euclidean space from the perspective of computational complexity. Let denote -dimensional hyperbolic space of sectional curvature . In hyperbolic space, the radius of the disks or balls matters; for example one gets different graph classes for radius 1 and radius 2 disks in . Hence, we parameterize the graph class of equal-sized balls by the radius of these balls, and denote the class by . We sometimes refer to graphs from a class as uniform ball graphs.
There have been several papers studying Independent Set, Dominating Set, Hamiltonian Cycle, -Coloring, etc. in unit ball graphs in Euclidean space [23, 24, 29, 2, 17, 13], all concluding that is the optimal running time for these problems in . In this paper, we show that a similar phenomenon occurs in , but shifted by one dimension: the problems can be solved in time in , just as in ; in general, for constant , the optimal running time is for these problems in under ETH.
In we see a similar drop in complexity, which is perhaps even more intriguing: several problems that are NP-complete in unit disk graphs in can be solved in quasi-polynomial () time for uniform disk graphs in , or even in polynomial time in bounded degree uniform disk graphs. This is perhaps the most striking consequence of our treewidth bounds. We also identify two problems, namely -Coloring for constant and Hamiltonian Cycle that admit polynomial time algorithms even in case of unbounded degree.
1.1 Noisy uniform ball graphs
Before we can state our contributions, we first define the graph classes that we consider more formally. We state the definitions for arbitrary metric spaces, although our main interest will be in hyperbolic space. Let be a metric space and let be parameters with . A graph is a -noisy uniform ball graph if there is a function , such that for all pairs we have:
In particular, pairs of vertices where can either be connected or disconnected. We denote the class of graphs defined this way by .
This can be regarded as a generalization of a unit disk graph, as follows. Define and , and consider the (multi)set of balls centered at points in and of radius . In a normal ball graph we would connect two nodes if and only if their balls intersect. In the class however, the boundaries of the balls are noisy, and nodes at distance in the range may or may not be connected. Although the class seems like a slight generalization of , it corresponds to a much larger graph class; this is shown in Theorem A in Appendix A.
The function is called an embedding of ; note that is not necessarily injective, so its image is a multiset. It is easy to see that an intersection graph of similarly sized fat objects with inscribed and circumscribed ball radii and is a noisy unit ball graph with and (therefore noisy uniform ball graphs are a direct generalization of intersection graphs of similarly sized fat objects as seen in ).
We are also interested in noisy unit ball graphs that are shallow, as defined next. The ply of a set of objects is the size of the largest subset of with a non-empty common intersection, i.e., . It will be useful to define a notion of sparseness that can be used for noisy unit ball graphs. The -ply of a point set is the ply of the set of balls of radius around points of . We say that a point set is -shallow if its -ply is at most . We denote by the set of graphs in that have a -shallow embedding. Note that for any fixed and , the shallow class is a very small subset of noisy unit ball graphs by Theorem A.
1.2 Our contribution
Our first contribution is a separator theorem that is inspired by the work of Fu .
Let , and and be a constants, and let be a -noisy unit ball graph. Then has a clique decomposition and a corresponding -balanced clique-separator of size and weight if , and size and weight if . Given and an embedding into , the clique decomposition and the separator can be found with a Las Vegas algorithm in expected time.
|Graph class||Indep. Set & others||-Coloring||Hamil. Cycle|
In case of , we can apply the standard way of turning balanced separators into a bound on treewidth, yielding tight treewidth bounds. When the size of the separator is logarithmic, however, one would get an extra logarithmic factor in the treewidth bound. This is significant because the treewidth typically shows up in the exponent of the running time, which means that the extra logarithmic factor can make the difference between polynomial and quasi-polynomial time. Section 3 and Theorem 3.2 shows that the extra logarithmic factor in the treewidth can be avoided. This is our second main contribution.
With the most important properties established, we can use techniques of De Berg et al. , as detailed in Section 3.3 to make our results more practical from the perspective of algorithms. We show that tree decompositions of width asymptotically matching our bounds can be found in these graph classes even in the absence of a geometric embedding, thereby making most of our algorithms embedding-free in the sense that they can work with only a graph as input. This requires that we work with a greedy partition, a partition that has a class for each vertex of a maximal independent set of a graph , where other vertices of are assigned greedily into a class of one of their neighbors from . The framework allows us to get Theorem 1.2, which serves as a basis for our algorithms. Note that this approach avoids the separator finding of Theorem 1.2, and it is deterministic.
Let and be fixed constants, and let with unknown embedding. Then for any greedy partition the -flattened treewidth of is in case of or if , and a weighted tree decomposition of of width (respectively, ) can be computed in (resp. ) time. If additionally , then the treewidth of is (resp. ), and a tree decomposition of width (resp. ) can be computed in polynomial (resp. ) time.
Section 4 showcases the algorithmic applications, as summarized in Table 1. Finally, we complement the quasi-polynomial algorithms in with a quasi-polynomial lower bound for Independent Set in under ETH. In this section, we also show that the lower bound framework of  for applies in . Consequently, the algorithms given in are also optimal up to constant factors in the exponent under ETH.
2 A separator theorem
We begin with some technical lemmas. A tiling of a is a set of interior disjoint compact subsets of that together cover ; in this paper, we only use tiles that are homeomorphic to a closed unit ball. We define tilings where each tile contains a ball of radius and contained in a ball of radius ; we call such a tiling a -nice tiling, or just nice tiling for short. A pair of distinct tiles are neighboring if their boundaries intersect in more than one point.
For any , there is a nice tiling of with isometric tiles of diameter where the distance between pairs of non-edge-neighboring tiles is .
For any there is a nice tiling of with isometric tiles of diameter .
Proof of Lemma 2.
(i) Let ; we define as the tiling of by regular -gons where at each vertex three -gons meet (that is, the regular tiling with Schläfli symbol ). We examine the right triangle created by the center of a tile, a midpoint of a side and a neighboring vertex . The triangle has angle at , since there are six such angles at , and it has angle at . From the hyperbolic law of cosines for angles () we have that the hypotenuse has length
where the second inequality uses that , which follows from the Laurent-series of around . It follows that the diameter of the tiles is . It is useful to find the lengths of the other two sides as well: they are and . The distance of non-neighboring cells is at least , and the radii of the inscribed and circumscribed circle of a tile are and respectively, each of which has length , therefore the tiling is nice.
(ii) We describe the shapes using the Poincaré half-space model of . The tiling is based on the tiling of Cannon et al. in Section 14 of . Let be the axis-aligned Euclidean hypercube with lexicographically minimal corner and hyperbolic diameter ; let be its Euclidean side length. (Note that this shape does not resemble a hypercube in any sense in .) See Figure 1. The hyperbolic diameter of this shape is realized for two opposing vertices of the hypercube, e.g. for the vertices and , so the diameter is
We note that for , we have , and otherwise . Then we can define as the image of for the following set of Euclidean transformations:
where and . Note that is an isometry since it can be regarded as the succession of a translation and a homothety from the origin, both of which are isometries of the half-space model. Finally, a Euclidean hypercube has a Euclidean inscribed ball, which is also an inscribed ball in the hyperbolic sense as well, although with different center and radius. A straightforward calculation gives that the hyperbolic radius of this ball is . In case of , this is , and otherwise . On the other hand, the tiles have diameter , therefore they have a circumscribed ball of radius , so the tiling is nice. ∎
Given a metric space and a subset , let the -neighborhood of is the set . Given a set of tiles , the neighborhood graph of is the graph with vertex set where tiles are connected if and only if they are neighbors. We denote the neighborhood graph by , and its shortest-path distance is denoted by , giving the metric space . In case of a tile set , let .
Since non-edge-neighboring tiles of from part (i) of Lemma 2 are at distance at least , a pair of tiles at -distance define an edge of . We will need this corollary later on.
For each there exists a constant such that for any set of tiles , the tiles in and their neighboring tiles cover the -neighborhood of , that is,
Our key lemma will rely on some basic properties of , namely that it is locally Euclidean and that the volume of a ball or sphere of radius is . The properties are explored in more detail in the following observation.
The following claims hold for any -nice tiling of .
The number of tiles contained in a radius ball is .
Let be the set of tiles whose distance from the origin is in the interval . Then .
Let be a uniform random hyperplane passing through the origin, and let be the set of points at distance from . Then a tile of is intersected by with probability .
(i) The volume of a ball of radius is
where depends only on , therefore for . We denote by the ball in with center and radius , and let be the origin. The number of tiles intersected by is at least , since the circumscribed balls of the intersected tiles cover . Each tile has diameter at most , so tiles contained in necessarily cover . Therefore, the number of tiles contained in is at least . For the lower bound, notice that inscribed balls of the contained tiles are interior disjoint, so there cannot be more than tiles contained in .
(ii) Consider the set of tiles whose distance from the origin is in the interval . These tiles are all contained in the set , whose volume is
therefore . For a lower bound, let be the set of balls of radius drawn around each of these tiles. Since this is a tiling of , their union covers the sphere of radius . Each ball in can cover at most total (-dimensional) volume of this sphere, whose total volume is
(iii) Let be a tile intersected by . Let be a point of realizing the distance to the origin, i.e., . Since , there is a point at distance at most from , which in turn is at most distance away from a point . It follows that . Since , its distance from a point in is at most ; let be such a point. Let be the cone touching , and let be its half-angle. Considering the Poincaré ball model centered at now, the hyperplane intersects the ball if and only if its normal is in the angular neighborhood of a great circle of the unit -sphere, i.e., the normal of is in , where the distance on is the geodesic metric inherited from . In the -flat given by and an arbitrary point where the line from touches , the triangle has a right-angle at , , , therefore we have
The angular neighborhood of a great circle in has volume
where is the volume of the -dimensional unit sphere. Therefore the probability that intersects a given tile from is ∎
The centerpoint of a point set is a point such that any hyperplane through has the property that both of its sides contain at most points from . A centerpoint in can be found using the algorithm created for .
Given a point set of size in , there exists a point such that for any hyperplane through , the two open half-spaces with boundary both contain at most points of . Furthermore, such a point can be computed using a randomized algorithm in expected time.
Let be the point set corresponding to in the Beltrami-Klein model of , that is, is a point set in the unit ball centered at the origin in . Let be the centerpoint of in the traditional (Euclidean) sense. It follows that any hyperplane through is a -balanced separator of , i.e., on both sides of there are at most points from . Consequently, the same holds in . We can convert from to in linear time, where the Euclidean coordinates of can be computed using the Euclidean algorithm in polynomial expected time . ∎
Let be a set of points in , and let be a centerpoint. Let be a nice tiling of . Fix a constant , and let be the set of tiles intersected by the -neighborhood of a hyperplane through chosen uniformly at random (i.e., having a normal chosen uniformly at random from ). Then the expected size of is for and for . The expected weight is for , and otherwise.
Proof of Lemma 2.
Throughout this proof, the constants may depend on . Notice that the expected value of depends only on the distribution of points over the tiles. Define the set of tiles . We have the following equation.
By Observation 2 (iii), if , then for some constant , thus
After this point, we concentrate on maximizing the function on the right hand side of our latest inequality; this will not necessarily maximize the expected value itself.
Note that is a decreasing function of . Suppose that and . The sum on the right hand side of (2) is increased if we swap the content of and . Let be the largest distance where contains a non-empty tile. Notice that if , then all tiles in the sets are non-empty. Since we have a ball of radius that contains only non-empty tiles and at most points, this gives an upper bound on by Observation 2 (i):
Furthermore, if we fix , the value of is maximized if the values are equal (i.e., we now allow fractional values fro ). From now on, assume that . The number of tiles in is between and by Observation 2 (ii). Therefore if with , then we have . Hence,
Observe that the expected size of can follow the same calculation, without the logarithmic term. In case of , we thus have
by inequality (3). For the weight bound in case of , we know that the weight of each tile is at most , and , therefore .
If , then the size can be bounded the following way.
We now bound the weight from (4) in case of . Let be a small positive constant. We upper bound by , and by (notice that the latter uses ).
Using inequality (3), we get
Proof of Theorem 1.2.
Let be our input graph with embedding , and let . We begin by computing a centerpoint of in expected time according to Lemma 2. Let be the resulting point. (Note that generally .)
Next, we fix a tiling with tile diameter for some small positive constant , so that any pair of points in the same tile are necessarily connected; this tiling can be constructed using Lemma 2 (ii). Let be the vertex partition of corresponding to this tiling, i.e., for each tile we create a partition class in , which necessarily forms a clique in . Let be a uniform random hyperplane through , and let be its -neighborhood. Lemma 2 shows that the set of cliques given by tiles that intersect has expected size and weight as desired. Due to the properties of a centerpoint, we have that on both sides of there are at most points. Moreover, the set is a separator: assume for contradiction that there is a pair on different sides of that are connected. Then the geodesic has a unique intersection with , and since , both and are longer than , therefore the distance of , so they cannot be connected; this is a contradiction.
By Markov’s inequality, with probability a random separator will have at most twice the expected weight, and similarly with probability it has at most twice the expected size. In fact, in case of having weight guarantees that the size is at most ; for , having size guarantees that the weight is . We can compute the size and weight of a separator in polynomial time. Consequently, we can find a separator of the desired size and weight in expected time. ∎
3 Treewidth bounds
3.1 Partitions and treewidth basics
Let be a simple graph, with some geometric embedding that may or may not be given with the input. A clique-partition of is a partition of where each partition class forms a clique in . A -partition is a partition of where each partition class induces a connected subgraph of that can be covered by cliques. For a graph , let be a maximal independent set. We create a partition class for each , and assign vertices in to the class of an arbitrary neighbor inside . A partition created this way is called a greedy partition. Note that greedy partitions can be found in polynomial time in any graph. For example, greedy partitions are -partitions in constant-dimensional Euclidean unit ball graphs .
The -contraction of is the graph obtained by contracting all edges induced by each partition class, and removing parallel edges; it is denoted by . The weight of a partition class is defined as . Given a set , its weight is defined as the sum of the class weights within, i.e., . Note that the weights of the partition classes define vertex weights in the contracted graph .
A tree decomposition of a graph is a pair where is a tree and is a mapping from the vertices of to subsets of called bags, with the following properties. Let be the set of bags associated to the vertices of . Then we have: (1) For any vertex there is at least one bag in containing it. (2) For any edge there is at least one bag in containing both and . (3) For any vertex the collection of bags in containing forms a subtree of . The width of a tree decomposition is the size of its largest bag minus 1, and the treewidth of a graph equals the minimum width of a tree decomposition of . We talk about pathwidth and path decomposition if is a path.
We will need the notion of weighted treewidth . Here each vertex has a weight, and the weighted width of a tree decomposition is the maximum over the bags of the sum of the weights of the vertices in the bag (note: without the ). The weighted treewidth of a graph is the minimum weighted width over its tree decompositions.
Now let be a partition of a given graph . We apply the concept of weighted treewidth to , where we assign each vertex of the weight , and refer to this weighting whenever we talk about the weighted treewidth of a contraction .
3.2 Treewidth in () and in
The size bound of our separator theorem can be used to get a bound on the pathwidth (and treewidth) of shallow graphs, since in a shallow graph each tile has points, and the weight of each non-empty tile is . Thus the bounds that we proved in the previous section for clique-based separators immediately give the same asymptotic bounds for normal separators in shallow graphs. To turn these bounds into bounds on treewidth, we only need Theorem 20 of , which yields the following corollary:
Fix the constants , and . Then for any on vertices the pathwidth is and if , then for any the pathwidth is .
Following techniques of  one can derive the bound for the -flattened treewidth of graphs in as well.
Although our bound on the weight of the separator in in Theorem 1.2 is optimal up to constant factors (it is attained for the input where each tile in a hyperbolic disk of radius contains points), the bound for in Corollary 3.2 is far from optimal. We could use the above separator theorem to directly design a divide and conquer algorithm for Independent Set, and the running time would be in , or in shallow graphs; both of these running times can be significantly improved by a better bound on treewidth.
Let be fixed constants, and let with a given embedding. Then there exists a clique-partition such that the -flattened treewidth of is . If in addition , then the treewidth of is .
In order to prove Theorem 3.2, we focus first on its second statement about shallow graphs. The next lemma (with ) allows us to transfer from shallow to non-shallow graphs.
Let with some fixed embedding, and let be the partition given by a -nice tiling of . Then .
We can assign an arbitrary point of inside tile to represent the corresponding vertex of ; let be such a function. Clearly any pair of vertices at distance less than are connected. Suppose that a pair of points and in tiles and have distance at least . Any point in is at most distance away from , that is, , and similarly . Since , any pair of points in and must have distance at least , i.e., any pair of points in and are not connected, so the corresponding vertices in are also not connected.
To prove that this is a shallow embedding, suppose that there is a point such that is in the intersection of more than balls of radius centered at points of . It follows that . It follows that covers more than tiles. Since the balls of radius within each tile are disjoint, there can be at most tiles inside, which is a contradiction. ∎
The weighted treewidth of is at most times its treewidth (since each node in has weight at most ), so Lemmas 2, 3.2 and the second part of Theorem 3.2 about shallow graphs together imply the first part of Theorem 3.2. The rest of this section is dedicated to the proof of the second part of Theorem 3.2. The proof relies crucially on the isoperimetric inequality , which states that for a simple closed curve of length bounding an area in the hyperbolic plane , we have where equality is attained only for a geodesic circle. In fact, we only need the following simple corollary.
All simple closed curves in of length bounding area satisfy .
Note that for a tile set from Lemma 2(ii), the neighborhood graph is planar. A hole for a set of tiles is a finite set such that is a closed curve and .
An outerplanar or -outerplanar graph is a planar graph which has a plane embedding where all vertices lie on the outer face. A -outerplanar graph is a planar graph which has an embedding where removing vertices of the outer face leads to a -outerplanar graph.
The neighborhood graph of any finite tile set is -outerplanar.
Let and be the area and circumference of a tile in respectively, and let be the neighborhood graph of our tile set . Let be the tile set obtained from by filling all of its holes with tiles. We use induction on : we claim that is -outerplanar. To prove this claim, we show that a constant proportion of all tiles are adjacent to the unbounded component of .
The boundary of is a collection of closed curves bounding interior disjoint regions whose union has precisely tiles. So by summing the inequality for each of these curves, we have that . Let be the set of tiles in adjacent to , i.e., the set of tiles in adjacent to a tile of the unbounded component of . The length of is at most , so we have that , hence . Clearly , so , as required.
The neighborhood graph of has outerplanarity at most by induction. Therefore the outerplanarity of is at most