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Embedding -Outerplanar Graphs Into ½

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Embedding -Outerplanar Graphs Into ½ k ` Embedding -Outerplanar Graphs into ½ Ý Þ Þ Ü Chandra Chekuri£ Anupam Gupta Ilan Newman Yuri Rabinovich Alistair Sinclair Abstract nal metric into a nice metric, solve the problem for the nice metric, and finally translate the solution back to the original We show that the shortest-path metric of any -outerplanar metric. graph, for any fixed , can be approximated by a probability distribution over tree metrics with constant distortion, and When the optimization problem is monotone and scal- able in the associated metric (as is usually the case), it is nat- hence also embedded into ½ with constant distortion. These graphs play a central role in polynomial time approximation ural to restrict one’s attention to nice metrics which dominate schemes for many NP-hard optimization problems on gen- the original metric, i.e., in which no distances are decreased. The maximum factor by which distances are stretched in the ¢ Ò eral planar graphs, and include the family of weighted planar grids. approximating metric is called the distortion of the embed- This result implies a constant upper bound on the ratio ding. Typically, the distortion translates more or less directly between the sparsest cut and the maximum concurrent flow into the approximation factor that one has to pay in trans- forming the problem from one metric to the other, so obvi- in multicommodity networks for -outerplanar graphs, thus extending a classical theorem of Okamura and Seymour [26] ously we seek an embedding with low distortion. The num- for outerplanar graphs, and of Gupta et al. [17] for treewidth- ber of applications of this paradigm has exploded in the past 2 graphs. In addition, we obtain improved approximation few years, and it has become a versatile and standard part of the algorithm designer’s toolkit: see the recent survey by In- ratios for -outerplanar graphs on various problems for which approximation algorithms are based on probabilistic dyk [19] and the forthcoming book by Matouˇsek [23, Chap- tree embeddings. We also conjecture that our random tree ter 10]. These applications have also given impetus to the study of the underlying theory of finite metric spaces. embeddings for -outerplanar graphs can serve as a building In this paper we will be concerned with embedding fi- block for more powerful ½ embeddings in future. nite metric spaces into ½ , i.e., real space endowed with the 1 Introduction ½ (or Manhattan) metric. Low distortion embeddings into ½ have been recognized, along with embeddings into Eu- 1.1 Background Many optimization problems on graphs ½ clidean space ¾ and into low-dimensional , to be of fun- and related combinatorial objects involve some finite metric damental importance in applications of the above paradigm, associated with the object. In particular, the shortest-path as well as for the underlying theory. One of several com- metric on the vertices of an undirected graph with nonnega- pelling reasons for studying ½ -embeddingscomes from their tive weights on the edges frequently plays an important role. intimate connection with the maxflow-mincut ratio in a mul- While for general metric spaces such an optimization prob- ticommodity flow network. Namely, if every shortest-path lem can be intractable, it is often possible to identify a subset metric on a given graph with arbitrary edge lengths can be of “nice” metricsfor which the problemis easy. Thus, a natu- « embedded into ½ with distortion at most , then the ratio be- ral approach to such problems — and one which has proved tween the sparsest cut and the maximum concurrent flow for highly successful in many cases — is to embed the origi- any set of capacities and demands on the graph is bounded 1 by « [22, 2, 17] . For more details on the sparsest cut prob- £ Lucent Bell Labs, 600 Mountain Avenue, Murray Hill NJ 07974. lem, its relation to embeddings, and its application to the Email: [email protected] design of a host of divide-and-conquer algorithms, see the Ý Lucent Bell Labs, 600 Mountain Avenue, Murray Hill NJ 07974. survey by Shmoys [31]. Email: [email protected]. Part of this work was Equally important in algorithmic applications are cer- done when the author was visiting the University of California, Berkeley, supported by a US-Israeli BSF grant #19999325. tain special ½ embeddings known as embeddings into ran- Þ Computer Science Department, University of Haifa, Haifa 31905, dom (dominating) trees, whereby the given metric is approx- g Israel. Email: filan, yuri @cs.haifa.ac.il. Supported in part imated by a probability distribution over tree metrics. Since by a US-Israeli BSF grant #19999325. every tree metric can be embedded isometrically (i.e., ex- Ü Computer Science Division, University of California, Berkeley CA 94720-1776. Email: [email protected]. Supported in part 1 ` by NSF grants CCR-9820951 and CCR-0121555 and by a US-Israeli BSF Indeed, the distortion of an optimal ½ embedding is exactly the worst grant #19999325. cut-flow ratio for any choice of capacities and demands. actly, or with distortion 1) into ½ , approximating a metric by constant distortion strictly larger than 1. (For example, the « à ¿ random trees with expected distortion immediately yields graph ¾ has treewidth 2 but is not isometrically embed- « an embedding into ½ with distortion . As has been recog- dable; see [12, Example 6.3.2] for a simple proof of this nized in the work of Bartal and others [1, 5, 6], random tree fact.) embeddings have many additional applications to online and Some, but not all of the above results carry over to approximation algorithms that are not enjoyed by arbitrary the more restrictive setting of embedding into random trees. ½ embeddings. In [17] it is shown how to embed outerplanar graphs into For general metrics the question of embeddability random trees with small constant distortion (note that the into ½ is essentially resolved: Bourgain [9] showed that isometric embedding of Okamura and Seymour is not a tree Ò Ç ´ÐÓ Òµ any -point metric can be embedded into ½ with embedding); on the other hand, in the same paper it is shown Òµ distortion, and a matching lower bound was established for that even series-parallel graphs incur a distortion ª´ÐÓ the shortest-path metrics of unit-weighted expander graphs for tree embeddings. Despite this limitation, it is worth in [22]. For embeddings into random trees, a construction pointing out that the random tree embeddings of outerplanar ´ÐÓ Ò ÐÓ ÐÓ Òµ of Bartal [6] yields a distortion of Ç for an graphs played a key role in the development of constant Ò arbitrary -point metric. distortion ½ embeddings of series-parallel graphs in [17]: However, tight bounds are still not known for many im- the trick was to combine the special structure of the tree portant classes of graphs, including planar graphs and graphs embeddings with judicious use of random cuts. with bounded treewidth; many such restricted classes are conjectured to be embeddable with constant distortion. In- 1.2 Results In this paper, we extend the above line of deed, the general question of how the topology of a graph research to a much wider class of planar graphs, namely affects its embeddability into ½ , and into random trees, is -outerplanar graphs for arbitary constant . Informally, one of the most important open issues in the area of met- a planar graph is -outerplanar if it has an embedding ric embeddings. (See, e.g., the tutorial by Indyk in the with disjoint cycles properly nested at most deep. A last FOCS [19].) In addition to its inherent mathematical in- formal definition is given in Section 2, while Figure 4.2 terest, this question impacts the design of approximation al- shows a simple example; a canonical example of a - ¢ Ò gorithms for many problems on restricted families of graphs outerplanar family is the sequence of rectangular and networks. grids. -outerplanar graphs play a central role in polynomial Some limited but interesting progress has been made on time approximation schemes for many NP-hard optimization embedding restricted metrics into ½ . Recently, Rao [27] problems on general planar graphs (see, e.g., [4]). Our main showed that the shortest-path metric of any graph that result is the following: à ½ excludes a ÖÖ is embeddable into with distortion Ô ¿ ´Ö ÐÓ Òµ ª´ÐÓ Òµ Ç . This beats the lower bound for gen- Ô THEOREM 1.1. Any shortest-path metric of a -outerplanar Ç ´ ÐÓ Òµ eral graphs for any constant Ö , and also gives dis- graph can be embedded into a probability distribution over tortion embeddings for the classes of planar and bounded- Ç ´ µ trees, and hence into ½ , with distortion for some treewidth graphs. However, Rao’s approach (of first embed- absolute constant . Moreover, such an embedding can be ding these graphs into ¾ and then using isometric embed- found in randomized polynomial time. ½ dings of ¾ into ) was shown to be tight in [24], where a Ô ÐÓ Òµ lower bound of ª´ distortion was shown for embed- Thus, not only do such graphs embed well into ½ , ding even treewidth-2 (and hence also planar) graphs into ¾ . but they even embed well into dominating trees. This is Approaching the question from the other direction, a Òµ in contrast to the lower bound of ª´ÐÓ for treewidth-2 celebrated theorem of Okamura and Seymour [26] implies graphs [17]. that any outerplanar metric can be embedded isometrically 2 Our result immediately implies a constant maxflow- into ½ . However, it has been shown that outerplanar graphs mincut ratio for arbitrary multicommodity flow problems on à are essentially the only graphs (with the exception of 4 ) -outerplanar graphs. This is the first progress in this di- that are isometrically embeddable into ½ [25].
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