In this paper we disprove the following conjecture due to Goemans [16] and Linial [24] (also see [5, 26]): “Every negative type metric embeds into ℓ1 with constant distortion.” We show that for every δ>0, and for large enough n, there is an n-point negative type metric which requires distortion at-least (log log n) 1/6−δ to embed into ℓ1. Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC) of Khot [19], establishing a previously unsuspected connection between PCPs and the theory of metric embeddings. We first prove that the UGC implies super-constant hardness results for (non-uniform) SPARSEST CUT and MINIMUM UNCUT problems. It is already known that the UGC also implies an optimal hardness result for MAXIMUM CUT [20]. Though these hardness results depend on the UGC, the integrality gap instances rely “only ” on the PCP reductions for the respective problems. Towards this, we first construct an integrality gap instance for a natural SDP relaxation of UNIQUE GAMES. Then, we “simulate ” the PCP reduction and “translate ” the integrality gap instance of UNIQUE GAMES to integrality gap instances for the respective cut problems! This enables us to prove a (log log n) 1/6−δ integrality gap for (non-uniform) SPARSEST CUT and MIN-IMUM UNCUT, and an optimal integrality gap for MAX-IMUM CUT. All our SDP solutions satisfy the so-called “triangle inequality ” constraints. This also shows, for the first time, that the triangle inequality constraints do not add any power to the Goemans-Williamson’s SDP relaxation of MAXIMUM CUT. The integrality gap for SPARSEST CUT immediately implies a lower bound for embedding negative type metrics into ℓ1. It also disproves the non-uniform version of Arora, Rao and Vazirani’s Conjecture [5], asserting that the integrality gap of the SPARSEST CUT SDP, with the triangle inequality constraints, is bounded from above by a constant.

We show that the MULTICUT, SPARSEST-CUT, and MIN-2CNF ≡ DELETION problems are NP-hard to approximate within every constant factor, assuming the Unique Games Conjecture of Khot [STOC, 2002]. A quantitatively stronger version of the conjecture implies inapproximability factor of Ω(log log n). 1.

Algorithms in varied fields use the idea of maintaining a distribution over a certain set and use the multiplicative update rule to iteratively change these weights. Their analysis are usually very similar and rely on an exponential potential function. We present a simple meta algorithm that unifies these disparate algorithms and drives them as simple instantiations of the meta algorithm. 1

We improve upon recent lower bounds on the minimum distortion of embedding certain finite metric spaces into L1. In particular, we show that for every n ≥ 1, there is an n-point metric space of negative type that requires a distortion of Ω(log log n) for such an embedding, implying the same lower bound on the integrality gap of a well-known semidefinite programming relaxation for sparsest cut. This result builds upon and improves the recent lower bound of (log log n) 1/6−o(1) due to Khot and Vishnoi [The unique games conjecture, integrality gap for cut problems and the embeddability of negative type metrics into l1, in Proceedings of the 46th Annual IEEE Symposium

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

Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, statistics, and mathematical psychology, to name a few. The theory of metric embedding received much attention in recent years by mathematicians as well as computer scientists and has been applied in many algorithmic applications. A cornerstone of the field is a celebrated theorem of Bourgain which states that every finite metric space on n points embeds in Euclidean space with O(log n) distortion. Bourgain’s result is best possible when considering the worst case distortion over all pairs of points in the metric space. Yet, it is possible that an embedding can do much better in terms of the average distortion. Indeed, in most practical applications of metric embedding the main criteria for the quality of an embedding is its average distortion over all pairs. In this paper we provide an embedding with constant average distortion for arbitrary metric spaces, while maintaining the same worst case bound provided by Bourgain’s theorem. In fact, our embedding possesses a much stronger property. We define the ℓq-distortion of a uniformly distributed pair of points. Our embedding achieves the best possible ℓq-distortion for all 1 ≤ q ≤ ∞ simultaneously. These results have several algorithmic implications, e.g. an O(1) approximation for the unweighted uncapacitated quadratic assignment problem. The results are based on novel embedding methods which improve on previous methods in another important aspect: the dimension. The dimension of an embedding is of very high importance in particular in applications and much effort has been invested in analyzing it. However, no previous result im-

Semidefinite programming (SDP) relaxations appear in many recent approximation algorithms but the only general technique for solving such SDP relaxations is via interior point methods. We use a Lagrangian-relaxation based technique (modified from the papers of Plotkin, Shmoys, and Tardos (PST), and Klein and Lu) to derive faster algorithms for approximately solving several families of SDP relaxations. The algorithms are based upon some improvements to the PST ideas — which lead to new results even for their framework — as well as improvements in approximate eigenvalue computations by using random sampling. 1.

Hierarchical graph decompositions play an important role in the design of approximation and online algorithms for graph problems. This is mainly due to the fact that the results concerning the approximation of metric spaces by tree metrics (e.g. [10, 11, 14, 16]) depend on hierarchical graph decompositions. In this line of work a probability distribution over tree graphs is constructed from a given input graph, in such a way that the tree distances closely resemble the distances in the original graph. This allows it, to solve many problems with a distance-based cost function on trees, and then transfer the tree solution to general undirected graphs with only a logarithmic loss in the performance guarantee. The results about oblivious routing [30, 22] in general undirected graphs are based on hierarchical decompositions of a different type in the sense that they are aiming to approximate the bottlenecks in the network (instead of the point-to-point distances). We call such decompositions cutbased decompositions. It has been shown that they also can be used to design approximation and online algorithms for a wide variety of different problems, but at the current state of the art the performance guarantee goes down by an O(log 2 n log log n)-factor when making the transition from tree networks to general graphs. In this paper we show how to construct cut-based decompositions that only result in a logarithmic loss in performance, which is asymptotically optimal. Remarkably, one major ingredient of our proof is a distance-based decomposition scheme due to Fakcharoenphol, Rao and Talwar [16]. This shows an interesting relationship between these seemingly different decomposition techniques. The main applications of the new decomposition are an optimal O(log n)-competitive algorithm for oblivious routing in general undirected graphs, and an O(log n)-approximation for Minimum Bisection, which improves the O(log 1.5 n) approximation

A common approach for solving computational problems over a difficult metric space is to embed the “hard ” metric into L1, which admits efficient algorithms and is thus considered an “easy ” metric. This approach has proved successful or partially successful for important spaces such as the edit distance, but it also has inherent limitations: it is provably impossible to go below certain approximation for some metrics. We propose a new approach, of embedding the difficult space into richer host spaces, namely iterated products of standard spaces like ℓ1 and ℓ∞. We show that this class is rich since it contains useful metric spaces with only a constant distortion, and, at the same time, it is tractable and admits efficient algorithms. Using this approach, we obtain for example the first nearest neighbor data structure with O(log log d) approximation for edit distance in nonrepetitive strings (the Ulam metric). This approximation is exponentially better than the lower bound for embedding into L1. Furthermore, we give constant factor approximation for two other computational problems. Along the way, we answer positively a question posed in [Ajtai, Jayram, Kumar, and Sivakumar, STOC 2002]. One of our algorithms has already found applications for smoothed edit distance over 0-1 strings [Andoni and Krauthgamer, ICALP 2008]. 1

In this paper we present a new approximation algorithm for Unique Games. For a Unique Game with n vertices and k states (labels), if a (1 − ε) fraction of all constraints is satisfiable, the algorithm finds an assignment satisfying a 1 − O(ε � log n log k) fraction of all constraints. To this end, we introduce new embedding techniques for rounding semidefinite relaxations of problems with large domain size. 1