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

The Johnson-Lindenstrauss Lemma shows that any n points in Euclidean space (with distances measured by the ℓ2 norm) may be mapped down to O((log n)/ε 2) dimensions such that no pairwise distance is distorted by more than a (1+ε) factor. Determining whether such dimension reduction is possible in ℓ1 has been an intriguing open question. We show strong lower bounds for general dimension reduction in ℓ1. We give an explicit family of n points in ℓ1 such that any embedding with distortion δ requires n Ω(1/δ2) dimensions. This proves that there is no analog of the Johnson-Lindenstrauss Lemma for ℓ1; in fact embedding with any constant distortion requires n Ω(1) dimensions. Further, embed-ding the points into ℓ1 with 1 + ε distortion requires n 1

We consider the problem of embedding finite metrics with slack: we seek to produce embeddings with small dimension and distortion while allowing a (small) constant fraction of all distances to be arbitrarily distorted. This definition is motivated by recent research in the networking community, which achieved striking empirical success at embedding Internet latencies with low distortion into low-dimensional Euclidean space, provided that some small slack is allowed. Answering an open question of Kleinberg, Slivkins, and Wexler [29], we show that provable guarantees of this type can in fact be achieved in general: any finite metric can be embedded, with constant slack and constant distortion, into constant-dimensional Euclidean space. We then show that there exist stronger embeddings into ℓ1 which exhibit

Motivated by applications in combinatorial optimization, we initiate a study of the extent to which the global properties of a metric space (especially, embeddability in ℓ1 with low distortion) are determined by the properties of small subspaces. We note connections to similar issues studied already in Ramsey theory, complexity theory (especially PCPs), and property testing. We prove both upper bounds and lower bounds on the distortion of embedding locally constrained metrics into various target spaces.

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In a typical instance of a network design problem, we are given a directed or undirected graph G = (V,E), non-negative edge-costs ce for all e ∈ E, and our goal is to find a minimum-cost subgraph H of G that satisfies some design criteria. For example, we may wish to find a minimum-cost set of edges that induces a connected graph (this is the minimum-cost spanning tree problem), or we might want to find a minimum-cost set of arcs in a directed graph such that every vertex can reach every other vertex (this is the minimum-cost strongly connected subgraph problem). This abstract model for network design problems has a large number of practical applications; the design process of telecommunication and traffic networks, and VLSI chip design are just two examples. Many practically relevant instances of network design problems are NP-hard, and thus likely intractable. This survey focuses on approximation algorithms as one possible way of circumventing this impasse. Approximation algorithms are efficient (i.e., they run in polynomial-time), and they compute solutions to a given instance of an optimization problem whose objective values are close to those of the respective optimum solutions. More concretely, most of the problems discussed in this survey are minimization problems. We then say that an algorithm is an α-approximation for a given problem if the ratio of the cost of an approximate solution computed by the algorithm to that of an optimum solution is at most α over all instances. In the

The Johnson-Lindenstrauss lemma shows that only d = O ( 1 ɛ 2 log n) dimensions are needed to embed any set of n points in L2 into ℓ d 2 with distortion at most (1 + ɛ). We will show that such dimension reduction is not possible in L1. In particular, we will give a set of n points in L1 that cannot be D-embedded into ℓ d 1 unless d ≥ n Ω(1/D2). This result was originally shown by Brinkman and Charikar [1], providing a negative answer to whether a Johnson-Lindenstrauss analog exists for L1, a previously open question (see e.g. [5]). Our lecture will follow a different proof by Lee and Naor [6]. The high-level overview of both proofs is simply to show that a particular point set cannot be embedded without the stated number of dimensions. The point set is the recursive diamond graph, which can be defined on n vertices, for n that is any power of 2. Values of n that are not powers of 2 are handled by noting that there exists a power of 2 such that the associated diamond graph is O(1)-equivalent to a point set of size n.

In recent years, considerable advances have been made in the study of properties of metric spaces in terms of their doubling dimension. This line of research has not only enhanced our understanding of finite metrics, but has also resulted in many algorithmic applications. However, we still do not understand the interaction between various graph-theoretic (topological) properties of graphs, and the doubling (geometric) properties of the shortest-path metrics induced by them. For instance, the following natural question suggests itself: given a finite doubling metric (V, d), is there always an unweighted graph (V ′ , E ′ ) with V ⊆ V ′ such that the shortest path metric d ′ on V ′ is still doubling, and which agrees with d on V. This is often useful, given that unweighted graphs are often easier to reason about. A first hurdle to answering this question is that subdividing edges can increase the doubling dimension unboundedly, and it is not difficult to show that the answer to the above question is negative. However, surprisingly, allowing a (1 + ε) distortion between d and d ′ enables us bypass this impossibility: we show that for any metric space (V, d), there is an unweighted graph (V ′ , E ′ ) with shortest-path metric d ′ : V ′ × V ′ → R≥0 such that • for all x, y ∈ V, the distances d(x, y) ≤ d ′ (x, y) ≤ (1 + ε) · d(x, y), and • the doubling dimension for d ′ is not much more than that of d, where this change depends only on ε and not on the size of the graph. We show a similar result when both (V, d) and (V ′ , E ′ ) are restricted to be trees: this gives a simpler proof that doubling trees embed into constant dimensional Euclidean space with constant distortion. We also show that our results are tight in terms of the tradeoff between distortion and dimension blowup.

Abstract. In this paper, we show how we can derive lower bounds and also compute the exact distortion for the line embeddings of some special metrics, especially trees and graphs with certain structure. Using linear programming to formulate a simpler version of the problem gives an interesting intuition and direction concerning the computation of general lower bounds for distortion into the line. We also show that our lower bounds on special cases of metrics are a lot better than previous lower bounds. 1