Optimal stochastic planarization
It has been shown by Indyk and Sidiropoulos [IS07] that any graph of genus can be stochastically embedded into a distribution over planar graphs with distortion . This bound was later improved to by Borradaile, Lee and Sidiropoulos [BLS09]. We give an embedding with distortion , which is asymptotically optimal.
Apart from the improved distortion, another advantage of our embedding is that it can be computed in polynomial time. In contrast, the algorithm of [BLS09] requires solving an NP-hard problem.
Our result implies in particular a reduction for a large class of geometric optimization problems from instances on genus- graphs, to corresponding ones on planar graphs, with a loss factor in the approximation guarantee.
Planar graphs constitute an important class of combinatorial structures, since they can be used to model a wide variety of natural objects. At the same time, they have properties that give rise to improved algorithmic solutions for numerous graph problems, if one restricts the set of possible inputs to planar graphs (see, for example [Bak94]).
One natural generalization of planarity involves the genus of a graph. Informally, a graph has genus , for some , if it can be drawn without any crossings on the surface of a sphere with additional handles (see Section 1.3). For example, a planar graph has genus , and a graph that can be drawn on a torus has genus at most .
In a way, the genus of a graph quantifies how far it is from being planar. Because of their similarities to planar graphs, graphs of small genus usually exhibit nice algorithmic properties. More precisely, algorithms for planar graphs can usually be extended to graphs of bounded genus, with a small loss in efficiency or quality of the solution (e.g. [CEN09]). Unfortunately, many such extensions are complicated and based on ad-hoc techniques.
Inspired by Bartal’s stochastic embedding of general metrics into trees [Bar96], Indyk and Sidiropoulos [IS07] showed that every metric on a graph of genus can be stochastically embedded into a planar graph with distortion at most exponential in (see Section 1.3 for a formal definition of stochastic embeddings). Since the distortion measures the ability of the probabilistic mapping to preserve metric properties of the original space, it is desirable to make this quantity as small as possible. The above bound was later improved by Borradaile, Lee, and Sidiropoulos [BLS09], who obtained an embedding with distortion polynomial in . In the present paper, we give an embedding with distortion , which matches the lower bound from [BLS09]. The statement of our main result follows.
Theorem 1.1 (Stochastic planarization).
Any graph of genus , admits a stochastic embedding into a distribution over planar graphs, with distortion . Moreover, given a drawing of into a genus- surface, the embedding can be computed in polynomial time.
We note that Theorem 1.1 can be equivalently stated for compact 2-dimensional simplicial manifolds, i.e. continuous spaces obtained by glueing together finitely many triangles, with every point having a neighborhood homeomorphic to a disk. The shortest-path metrics of genus- graphs, are precisely the metrics supported on the 0-simplices of such genus- manifolds. The result for these spaces can be obtained via a careful affine extension of our embedding over simplices. Since our focus is on algorithmic applications, we omit the details, and restrict our discussion to the discrete case (i.e. finite graphs).
1.1 Our techniques
In [IS07] it was shown that a graph of genus can be stochastically embedded into a distribution over graphs of genus , with constant distortion. Repeating this times results in a planar graph, but yields distortion exponential in . The improvement of [BLS09] was obtained by giving an algorithm that removes all handles at once. The main technical tool used to achieve this was the Peeling Lemma from [LS09]. The idea is that given a graph of genus , one can find a subgraph , which we refer to as the cut graph, such that (i) is planar, (ii) has dilation , and (iii) can be stochastically embedded into a planar graph. The resulting distortion of the embedding produced via the Peeling Lemma is proportional to the dilation of , and therefore polynomial in .
It was further shown in [BLS09] that any cut graph has dilation , imposing a limitation on their technique. We overcome this barrier as follows. We first find a cut graph consisting of shortest paths with a common end-point. These paths are obtained from the generators of the fundamental group of the underlying surface, due to Erickson and Whittlesey [EW05]. In the heart of our analysis, we show how to embed a collection of shortest paths with a common end-point, into a random tree with distortion . This result can be viewed as a generalization of the tree-embedding theorem due to Fakcharoenphol, Rao, and Talwar [FRT03], who showed that any -point metric space admits a stochastic embedding into a tree with distortion .
This connection with tree embeddings seems surprising, since planar graphs appear to be significantly more complicated topologically. For instance, even embedding the grid into a random tree, requires distortion , due to a lower bound of Alon, Karp, Peleg, and West [AKPW91]. Gupta, Newman, Rabinovich, and Sinclair [GNRS99] have shown that the same lower bound of holds even for embedding very simple classes of planar graphs into trees, such as series-parallel graphs (i.e. even for planar graphs of treewidth 2).
Our tree-embedding result is obtained by combining the approach from [FRT03] with the algorithm of Lee and Sidiropoulos [LS10] for computing random partitions for graphs of small genus.
We remark however that the algorithm of [FRT03] computes an embedding into an ultrametric
As in the case of stochastic embeddings of arbitrary metrics into trees [Bar96], we obtain a general reduction from a class of optimization problems on genus- graphs, to their restriction on planar graphs. We now state precisely the reduction. Let be a set, a set of non-negative vectors corresponding all feasible solutions for a minimization problem, and . Then, we define the linear minimization problem to be the computational problem where we are given a graph , and we are asked to find , minimizing
Observe that this definition captures a very general class of problems. For example, MST can be encoded by letting be the set of indicator vectors of the edges of all spanning trees on , and the all-ones vector. Similarly, one can easily encode problems such as TSP, Facility-Location, -Server, Bi-Chromatic Matching, etc.
We can now state an immediate Corollary of our embedding result.
Let be a linear minimization problem. If there exists a polynomial-time -approximation algorithm for on planar graphs, then there exists a randomized polynomial-time -approximation algorithm for on graphs of genus .
One of the most intriguing open problems in the theory of metric embeddings is determining the optimal distortion for embedding planar graphs, and more generally graphs that exclude a fixed minor, into (see e.g. [LLR94, GNRS99, CGN03, LS09]). We remark that by the work of Linial, London, and Rabinovich [LLR94], this distortion equals precisely the maximum multi-commodity max-flow/min-cut gap on these graphs, and is therefore of central importance in divide-and-conquer algorithms that are based on Sparsest-Cut [LR99, ARV04]. Our embedding result immediately implies the following corollary. The first proof of this statement was given in [LS10], where it was derived via a fairly complicated argument.
If all planar graphs embed into with distortion at most , then all graphs of genus embed into with distortion .
Throughout the paper, we consider graphs with a non-negative length function . For a pair , we denote the length of the shortest path between and in , with the lengths of edges given by , by . Unless otherwise stated, we restrict our attention to finite graphs.
Graphs on surfaces
Let us recall some notions from topological graph theory (an in-depth exposition can be found in [MT01]). A surface is a compact connected 2-dimensional manifold, without boundary. For a graph we can define a one-dimensional simplicial complex associated with as follows: The -cells of are the vertices of , and for each edge of , there is a -cell in connecting and . A drawing of on a surface is a continuous injection . The orientable genus of a graph is the smallest integer such that can be drawn into a sphere with handles. Note that a graph of genus is a planar graph.
A mapping between two metric spaces and is non-contracting if for all . If is any finite metric space, and is a family of finite metric spaces, we say that admits a stochastic -embedding into if there exists a random metric space and a random non-contracting mapping such that for every ,
The rest of the paper is organized as follows. In Section 2 we show that in any graph of genus , we can find a collection of shortest paths with a common end-point, whose removal leaves a planar graph. In Section 3 we define alternating partitions for the metric space induced on these paths. Using these partitions, we show in Section 4 how to embed into a random tree, with distortion . Finally, in Section 5 we combine this tree embedding with the Peeling Lemma, to obtain our main result.
2 Homotopy generators
Let be a genus- graph embedded into an orientable genus- surface , and let be a vertex of . A system of loops with basepoint is a collection of cycles containing , such that the complement of in is homeomorphic to a disk. Examples of systems of loops are depicted in figure 1 (see also [EW05] for a detailed exposition). The set of cycles in a system of loops generate the fundamental group .
A system of loops is called optimal if every is the shortest cycle in its homotopy class. Algorithms for computing optimal systems of loops have been given by Colin de Verdière and Lazarus [dVL02] and by Erickson and Whittlesey [EW05]. The later algorithm has the property that each cycle can be decomposed into either two shortest paths with common end-point , or two such shortest paths, and an edge between the other two end-points. We therefore have the following.
Lemma 2.1 (Greedy homotopy generators [Ew05]).
Let be a graph embedded into an orientable surface of genus . Then, there exists a subgraph of satisfying the following properties:
(i) The complement of in is homeomorphic to a disk.
(ii) There exists , and a collection of shortest-paths in , having as a common end-point, such that .
3 Alternating partitions
Let be a graph. By rescaling the edge-lengths we may assume w.l.o.g. that the minimum distance in is one. Let be a collection of shortest paths in , with a common end-point . Let . We consider the metric space . We define a collection , where each is a random partition of into sets of diameter less than , and such that for any , is a refinement of . We refer to the resulting collection as alternating partitions for .
Pick a permutation , and reals
We set , i.e. the trivial partition that places all points into the same cluster. For , given we define by performing two partitioning steps that we describe below (see also figure 2).
Horizontal partitioning step: Let . We partition into clusters . We consider the paths in in the order . For each , we form the cluster
where . We say that the path is the trunk of . For notational convenience, we also refer to as the trunk of the unique cluster in the partition . We refer to the clusters as the horizontal children of .
Vertical partitioning step: Next, we proceed to partition each horizontal child of into a set of clusters , so that for any integer ,
We refer to the clusters as the vertical children of . We also say that is the trunk of . Finally, we add all non-empty clusters to .
This concludes the description of the construction of the alternating partitions for .
For any , and , we have .
Let be the trunk of . Let be the subpath of that is contained in . By the construction of the vertical children we have . Moreover, by the construction of the horizontal children we have that for any , . Therefore, for any we have . ∎
4 Embedding the cut graph into a random tree
As in the previous section, let be a graph, and . Let be shortest paths in with common end-point , and define . We will use the alternating partitions constructed in the previous section to obtain a stochastic embedding of into a distribution over trees, with distortion .
For any , let , and .
We proceed by induction on the partitions , starting from . For every cluster we construct a tree and an injection . We inductively maintain the following invariant:
(I) For every cluster with trunk , there exists in a copy of the subpath of containing all vertices with . We refer to this path as the stem of . We denote by the vertex in the stem of which is closest to in . We refer to as the root of .
By Lemma 3.1 we have that every cluster has diameter less than the minimum distance in , and therefore contains a single vertex. We set to be the trivial tree containing that vertex. The map sends the unique vertex in to its copy in .
Suppose now that we have constructed a tree for every cluster in , for some . We will show how to obtain a tree for every cluster in . Let , and let be the horizontal children of . For a horizontal child , let be its vertical children. Recall that each such is a cluster in . Therefore, by the induction hypothesis we have already computed a tree for every , and an injection . We construct the tree in two steps:
Vertical composition step: We first combine the graphs of the vertical children of each , to obtain an intermediate tree . This is done as follows. Recall that is the trunk of . By the inductive invariant (I) we have that for every vertical child , its stem is a path in , and each such is a copy of a subpath of . In particular, since for all we have , it follows that the stems of distinct vertical children correspond to disjoint subpaths of . Let be the subpath containing all vertices with . Since it follows that . We form the tree by taking a copy of and identifying for every vertical child , the stem in with its copy in . The path becomes the stem of . The mapping is defined by composing each with the natural inclusion .
Horizontal composition step: Next, we combine the trees for all horizontal children of , to obtain . Let be the trunk of . Observe that there exists a non-empty horizontal child of . For any , with , we connect with via an edge of length . Let be the resulting tree, and be the induced injection from to . Note that the root of is .
It is easy to verity that the inductive invariant (I) is maintained. Finally, we set , and . This concludes the description of the embedding .
4.1 Bounding the distortion
It is straight-forward to verify that the mapping is non-contracting, so it remains to bound the expected expansion for every pair of vertices. We begin with a useful Lemma.
Let , let , and let be the stem of . Then, for any , we have .
For any , let . We have that is the tree containing only . For any , let be the stem of . We have , and . Thus ∎
For the remaining of the analysis, we fix two vertices . We wish to bound , where the expectation is taken over the randomness used in constructing the alternating partitions (i.e. , , and ).
We begin by introducing some notation. We say that a path settles at level if and are in the same cluster in , and is the first path w.r.to the ordering such that is the trunk of at least one of the clusters , . Moreover, we say that cuts horizontally at level if it settles at level , and exactly one of the clusters , has as its trunk.
Similarly, we say that saves at level if and are in the same cluster in , and is the trunk of a cluster in containing both and . We say that cuts vertically at level if it saves at level , there is a horizontal child of a cluster in containing both and , and .
Let , resp. , be the supremum of when cuts at level horizontally, resp. vertically, taken over all possible random choices of the algorithm. That is,
Then, we have
We will bound each one of these quantities separately.
Define the interval
In order for to cut horizontally at level it must be the case that . Since is chosen from uniformly at random, it follows by the triangle inequality that this happens with probability at most
Assume w.l.o.g. that . Conditioned on the event that , any of the paths can settle . Therefore,
Next we bound . Suppose that a path cuts horizontally at level . Let be the stem of the cluster in containing both and . By Lemma 4.1 we conclude that
Observe that since , it follows that for every , the path can cut only at a single level . Therefore
Denote by the event , and by the event . In order for to cut vertically at level , both and must hold.
Assume w.l.o.g. that . Conditioned on , any of can save . Therefore,
By the triangle inequality we have
We next upper bound . Suppose that cuts vertically at level . Let , and . Assume w.l.o.g. that . If follows by the construction of that is the trunk of and . Therefore, there exist clusters with , , such that is the trunk of every , and such that the stems of are consecutive subpaths of . Let be the stem of . For any , the path is connected to path via a path , such that . By lemma 4.1 we have
Let be a graph, and let be a collection of shortest-paths in , sharing a common end-point. Then, the metric space admits a stochastic embedding into a distribution over trees with distortion .
Let be a metric space. A distribution over partitions of is called -Lipschitz if every partition in the support of has only clusters of diameter at most , and for every ,
We denote by the infimum such that for any , the metric admits a -Lipschitz random partition, and we refer to as the modulus of decomposability of . The following theorem is due to Klein, Plotkin, and Rao [KPR93], and Rao [Rao99].
Let be a graph, and let . The dilation of is defined to be
For a graph and a graph family we write to denote the fact that stochastically embeds into a distribution over graphs in , with distortion .
For two graphs , a 1-sum of with is a graph obtained by taking two disjoint copies of and , and identifying a vertex with a vertex . For a graph family , we denote by the closure of under 1-sums.
Lemma 5.2 (Peeling Lemma [Ls09]).
Let be a graph, and . Let be a graph with , and let be the corresponding modulus of decomposability. Then, there exists a graph family such that , where , and every graph in is a 1-sum of isometric copies of the graphs and .
We will use the following auxiliary Lemma.
Lemma 5.3 (Composition Lemma).
Let be a graph, and let , , be graph families. If , and for any , , then .
It follows by direct composition of the two embeddings. ∎
Proof of Theorem 1.1.
Let be a genus- graph, drawn on a genus- surface . Let be the subgraph of given by Lemma 2.1. Recall that there exists and shortest paths in , having as a common end-point, such that .
Let us write , and . By Theorem 4.4, we have
After scaling the lengths of the edges in , we may assume that the minimum distance is one. Let be the graph obtained from as follows. For every edge with and , we replace with two edges and , with , and . Let , i.e. the set , together with all new vertices introduced above.
Observe that for any , we have . Therefore by (9),
This embedding can be extended to as follows. For every tree in the support of the distribution, and for every vertex , we attach to by adding an edge of length between and the unique neighbor of in . Since we only add leaves to , the new graph is still a tree. It is straight-forward to verify that the resulting stochastic embedding has distortion , and thus
Let be the graph obtained from by adding an edge of length , between every pair of vertices . Let , and . By cutting the surface along we obtain a drawing of into the interior of a disk. Since every vertex in is attached to via a single edge, it follows that this planar drawing can be extended to . Thus, the graph is planar, and by theorem 5.1 we have .
Similarly, we have that for any , the graph is planar.
Observe that . By the Peeling Lemma (Lemma 5.2) we have that can be stochastically embedded with distortion into a distribution over graphs , where is obtained by 1-sums of isometric copies of and . Each graph in is planar, and therefore
By (10), and since the metric is the same as the metric , we have that
Since contains an isometric copy of , this implies
concluding the proof. ∎
The author wishes to thank Glencora Borradaile, Piotr Indyk, and James R. Lee for their invaluable contributions to the development of the techniques used in this paper. He also thanks Dimitrios Thilikos for a question which motivated the problem studied here, as well as Jeff Erickson, and Sariel Har-Peled for several enlightening discussions.
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