Improved lower bounds for embeddings into L₁

Abstract

Abstract. We improve upon recent lower bounds on the minimum distortion of embedding certain finite metric spaces into L 1 . 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 1. Introduction. In recent years, low distortion embeddings of finite metric spaces into L 1 have become a powerful tool in an algorithm designer's arsenal. Such embeddings are extremely useful in two very different contexts, which we discuss below.In combinatorial optimization, cuts in an n-vertex graph correspond to n-point cut semimetrics (see, e.g., [5,16,28]).1 These semimetrics span the cone of n-point semimetrics that are subsets of L 1 . Polynomial-time computable relaxations of NPhard cut problems are often expressed as optimization over larger sets of semimetrics. Thus, using ("rounding") a relaxed solution to approximate the optimal solution to the original problem often boils down to embedding the relaxed solution into L 1 .In data analysis, proximity and classification problems are often easier to perform when data sets are subsets of L 1 . In fact, for many such problems L 1 behaves as well as Euclidean space (see, for example, [19,24]). Therefore, a common approach to solving such problems in other input families is first to embed the distance function into a well-behaved normed space such as L 1 , and then to use known solutions for the chosen target space.We consider in this paper the embedding into L 1 of two types of metrics that have attracted much attention recently. First, we consider negative type metrics.

Expander flows, geometric embeddings and graph partitioning

“…The hypercube example appears to be a serious obstacle to improving the approximation factor for finding sparse cuts, since the optimal cuts of this graph (namely, the dimension cuts) elude all our rounding techniques. In fact, hypercube-like examples have been used to show that this particular SDP approach has an integrality gap of Ω(log log n) (Devanur et al [2006;Krauthgamer and Rabani [2006]). …”

“…Originally it was suggested in Goemans [1998] that the SDP with triangle inequality has an integrality gap of O(1) but Khot and Vishnoi [2005] showed that the gap is Ω(log log 1/6 n), which was improved to Ω(log log n) by Krauthgamer and Rabani [2006] for the nonuniform sparsest cut. Recently, Devanur et al [2006] showed that the Ω(log log n) gap holds even for the uniform version.…”

Lower bounds for Edit Distance and Product Metrics via Poincaré-Type Inequalities

“…The question of (non-)embeddability of edit distance into 1 appears on the Matoušek's list of open problems [Mat07], as well as in Indyk's survey [Ind01]. From the first nonembeddability bound of 3/2 of [ADG + 03], the bound has been improved to Ω(log 0.5−o(1) d) by Khot and Naor [KN06], and later to the state-of-the-art Ω(log d) bound of Krauthgamer and Rabani [KR06]. Later, Andoni and Krauthgamer [AK07] prove an Ω( log d log log d ) lower bound for embedding into more general classes of spaces, which includes 1 .…”

On Khot’s unique games conjecture

“…This has been a rather unique approach to the construction of counterexamples in metric geometry. The lower bound was improved to Ω(log log n) by Krauthgamer and Rabani [27], and shortly afterward Devanur, Khot, Saket, and Vishnoi [18] showed that even the Sparsest Cut relaxation has an integrality gap Ω(log log n).…”