In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; there exist metric spaces where any tree embedding must have distortion#sto n)-distortion. This problem lies at the heart of numerous approximation and online algorithms including ones for group Steiner tree, metric labeling, buy-at-bulk network design and metrical task system. Our result improves the performance guarantees for all of these problems.

At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a clique-sum of pieces almost-embeddable into boundedgenus surfaces. This result has many applications. In particular, we show applications to developing many approximation algorithms, including a 2-approximation to graph coloring, constant-factor approximations to treewidth and the largest grid minor, combinatorial polylogarithmicapproximation to half-integral multicommodity flow, subexponential fixed-parameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor. 1.

In this paper, we study the metrics of negative type, which are metrics (V,d) such that √ d is an Euclidean metric; these metrics are thus also known as “ℓ2squared” metrics. We show how to embed n-point negative-type metrics into Euclidean space ℓ2 with distortion D = O(log 3/4 n). This embedding result, in turn, implies an O(log 3/4 k)-approximation algorithm for the Sparsest Cut problem with non-uniform demands. Another corollary we obtain is that n-point subsets of ℓ1 embed into ℓ2 with distortion O(log 3/4 n). 1

We introduce a new framework for designing fixed-parameter algorithms with subexponential running time---2 . Our results apply to a broad family of graph problems, called bidimensional problems, which includes many domination and covering problems such as vertex cover, feedback vertex set, minimum maximal matching, dominating set, edge dominating set, clique-transversal set, and many others restricted to bounded genus graphs. Furthermore, it is fairly straightforward to prove that a problem is bidimensional. In particular, our framework includes as special cases all previously known problems to have such subexponential algorithms. Previously, these algorithms applied to planar graphs, single-crossing-minor-free graphs, and/or map graphs; we extend these results to apply to bounded-genus graphs as well. In a parallel development of combinatorial results, we establish an upper bound on the treewidth (or branchwidth) of a bounded-genus graph that excludes some planar graph H as a minor. This bound depends linearly on the size (H)| of the excluded graph H and the genus g(G) of the graph G, and applies and extends the graph-minors work of Robertson and Seymour. Building on these results...

We solve an open problem posed by Eppstein in 1995 [14, 15] and re-enforced by Grohe [16, 17] concerning locally bounded treewidth in minor-closed families of graphs. A graph has bounded local treewidth if the subgraph induced by vertices within distance r of any vertex has treewidth bounded by a function of r (not n). Eppstein characterized minor-closed families of graphs with bounded local treewidth as precisely minor-closed families that minor-exclude an apex graph, where an apex graph has one vertex whose removal leaves a planar graph. In particular, Eppstein showed that all apex-minor-free graphs have bounded local treewidth, but his bound is doubly exponential in r, leaving open whether a tighter bound could be obtained. We improve this doubly exponential bound to a linear bound, which is optimal. In particular, any minor-closed graph family with bounded local treewidth has linear local treewidth. Our bound generalizes previously known linear bounds for special classes of graphs proved by several authors. As a consequence of our result, we obtain substantially faster polynomial-time approximation schemes for a broad class of problems in apex-minor-free graphs, improving the running time from .

It was shown recently by Fakcharoenphol et al. [9] that arbitrary finite metrics can be embedded into distributions over tree metrics with distortion O(log n). It is also known that this bound is tight since there are expander graphs which cannot be embedded into distributions over trees with better than Ω(log n) distortion. We show that this same lower bound holds for embeddings into distributions over any minor excluded family. Given a family of graphs F which excludes minor M where |M | = k, we explicitly construct a family of graphs with treewidth-(k + 1) which cannot be embedded into a distribution over F with better than Ω(log n) distortion. Thus, while these minor excluded families of graphs are more expressive than trees, they do not provide asymptotically better approximations in general. An important corollary of this is that graphs of treewidth-k cannot be embedded into distributions over graphs of treewidth-(k−3) with distortion less than Ω(log n). We also extend a result of Alon et al. [1] by showing that for any k, planar graphs cannot be embedded into distributions over treewidth-k graphs with better than Ω(log n) distortion. 1

Embedding between metric spaces is a very powerful algorithmic tool and has been used for finding good approximation algorithms for several problems. In particular, embedding to an ℓ1 norm has been used as the key step in an approximation algorithm for the sparsest cut problem. The sparsest cut problem, in turn, is the main ingredient of many algorithms that have a divide and conquer nature and are used in various fields. While every metric is embeddable into ℓ1 with distortion O(log n) [13], and the bound is tight [39], for special classes of metrics better bounds exist. Shortest path metrics for trees and outerplanar graphs are isometrically embeddable into ℓ1 [41]. Series-parallel graphs [28] and k-outerplanar graphs [19](for constant k) are embeddable into ℓ1 with constant distortion, planar graphs and bounded tree-width graphs are conjectured to have constant distortion in embedding to ℓ1. Bounded tree-width graphs are one of most general graph classes on which several hard problems are tractable. We study the embedding of series-parallel graphs (or, more generally, graphs

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Abstract. Gupta (SODA’01) considered the Steiner Point Removal (SPR) problem on trees. Given an edge-weighted tree T and a subset S of vertices called terminals in the tree, find an edge-weighted tree TS on the vertex set S such that the distortion of the distances between vertices in S is small. His algorithm guarantees that for any finite tree, the distortion incurred is at most 8. Moreover, a family of trees, where the leaves are the terminals, is presented such that the distortion incurred by any algorithm for SPR is at least 4(1 − o(1)). In this paper, we close the gap and show that the upper bound 8 is essentially tight. In particular, for complete binary trees in which all edges have unit weight, we show that the distortion incurred by any algorithm for the SPR problem must be at least 8(1 − o(1)). 1

Consider a routing problem instance consisting of a demand graph H = (V, E(H)) and a supply graph G = (V, E(G)). If the pair obeys the cut condition, then the flow-cut gap for this instance is the minimum value C such that there exists a feasible multiflow for H if each edge of G is given capacity C. It is wellknown that the flow-cut gap may be greater than 1 even in the case where G is the (series-parallel) graph K2,3. In this paper we are primarily interested in the “integer ” flow-cut gap. What is the minimum value C such that there exists a feasible integer valued multiflow for H if each edge of G is given capacity C? We formulate a conjecture that states that the integer flow-cut gap is quantitatively related to the fractional flow-cut gap. In particular this strengthens the well-known conjecture that the flow-cut gap in planar and minor-free graphs is O(1) [12] to suggest that the integer flow-cut gap is O(1). We give several technical tools and results on non-trivial special classes of graphs to give evidence for the conjecture and further explore the “primal ” method for understanding flow-cut gaps; this is in contrast to and orthogonal to the highly successful metric embeddings approach. Our results include the following: • Let G be obtained by series-parallel operations starting from an edge st, and consider orienting all edges in G in the direction from s to t. A demand is compliant if its endpoints are joined by a directed