The unique games conjecture, integrality gap for cut problems and embeddability of negative-type metrics into ℓ1

Subhash A. Khot, Nisheeth K. Vishnoi

Research output: Contribution to journalArticle

Abstract

In this article, we disprove a conjecture of Goemans and Linial; namely, that every negative type metric embeds into ℓ1 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 ℓ1 . 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 ofUNIQUEGAMES 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.

Original languageEnglish (US)
Number of pages1
JournalJournal of the ACM
Volume62
Issue number1
DOIs
StatePublished - Feb 1 2015

Keywords

  • Hardness of approximation
  • Integrality gap
  • Metric embeddings
  • Negative-type metrics
  • Semidefinite programming
  • Sparsest cut
  • Unique games conjecture

ASJC Scopus subject areas

  • Software
  • Control and Systems Engineering
  • Information Systems
  • Hardware and Architecture
  • Artificial Intelligence

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