### The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative-Type Metrics into $ℓ_{1}$The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative-Type Metrics into $ℓ_{1}$

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 Author Khot, Subhash A. ♦ Vishnoi, Nisheeth K. Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©2015 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Metric embeddings ♦ Hardness of approximation ♦ Integrality gap ♦ Negative-type metrics ♦ Semidefinite programming ♦ Sparsest cut ♦ Unique games conjecture Abstract 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 $\textit{n},$ there is an $\textit{n}-point$ 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 (nonuniform) 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. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2015-03-02 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 62 Issue Number 1 Page Count 39 Starting Page 1 Ending Page 39

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Source: ACM Digital Library