### 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 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 (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 language | English (US) |
---|---|

Pages (from-to) | 8 |

Number of pages | 1 |

Journal | Journal of the ACM |

Volume | 62 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2015 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

**The unique games conjecture, integrality gap for cut problems and embeddability of negative-type metrics into ℓ1
.** / Khot, Subhash; Vishnoi, Nisheeth K.

Research output: Contribution to journal › Article

*Journal of the ACM*, vol. 62, no. 1, pp. 8. https://doi.org/10.1145/2629614

}

TY - JOUR

T1 - The unique games conjecture, integrality gap for cut problems and embeddability of negative-type metrics into ℓ1

AU - Khot, Subhash

AU - Vishnoi, Nisheeth K.

PY - 2015/2/1

Y1 - 2015/2/1

N2 - 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.

AB - 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.

KW - Hardness of approximation

KW - Integrality gap

KW - Metric embeddings

KW - Negative-type metrics

KW - Semidefinite programming

KW - Sparsest cut

KW - Unique games conjecture

UR - http://www.scopus.com/inward/record.url?scp=84923924156&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84923924156&partnerID=8YFLogxK

U2 - 10.1145/2629614

DO - 10.1145/2629614

M3 - Article

AN - SCOPUS:84923924156

VL - 62

SP - 8

JO - Journal of the ACM

JF - Journal of the ACM

SN - 0004-5411

IS - 1

ER -