The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative-Type Metrics into ℓ_1

摘要

In this article, we disprove a conjecture of Goemans and Linial; namely, that every negative type metric embeds into ℓ_1 with constant distortion. We show that for an arbitrarily small constant δ ＞ 0, for all 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 ℓ_1. Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC), establishing a previously unsuspected connection between probabilistically checkable proof systems (PCPs) and the theory of metric embeddings. We first prove that the UGC implies a super-constant hardness result for the (nonuni-form) SparsestCut problem. Though this hardness result relies on the UGC, we demonstrate, nevertheless, that the corresponding PCP reduction can be used to construct an "integrality gap instance" for SparsestCut. Towards this, we first construct an integrality gap instance for a natural SDP relaxation of UniqueGames. Then we "simulate" the PCP reduction and "translate" the integrality gap instance of UniqueGames to an integrality gap instance of SparsestCut. This enables us to prove a (log log n)~(1/6-δ) integrality gap for SparsestCut, which is known to be equivalent to the metric embedding lower bound.