SDP integrality gaps with local ℓ1-embeddability

Subhash Khot, Rishi Saket

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We construct integrality gap instances for SDP relaxation of the MAXIMUM CUT and the SPARSEST CUT problems. If the triangle inequality constraints are added to the SDP, then the SDP vectors naturally define an n-point negative type metric where n is the number of vertices in the problem instance. Our gap-instances satisfy a stronger constraint that every sub-metric on t = O((log log log n)1/6) points is isometrically embeddable into ℓ1. The local ℓ1-embeddability constraints are implied when the basic SDP relaxation is augmented with t rounds of the Sherali-Adams LP-relaxation. For the MAXIMUM CUT problem, we obtain an optimal gap of αGW -1 - ε, where αGW is the Goemans-Williamson constant [11] and ε > 0 is an arbitrarily small constant. For the SPARSEST C UT problem, we obtain a gap of Ω((log log log n) 1/13). The latter result can be rephrased as a construction of an n-point negative type metric such that every t-point sub-metric is isometrically ℓ1 -embeddable, but embedding the whole metric into ℓ1 incurs distortion Ω((log log log n) 1/13).

Original languageEnglish (US)
Title of host publicationProceedings - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009
Pages565-574
Number of pages10
DOIs
StatePublished - Dec 1 2009
Event50th Annual Symposium on Foundations of Computer Science, FOCS 2009 - Atlanta, GA, United States
Duration: Oct 25 2009Oct 27 2009

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Other

Other50th Annual Symposium on Foundations of Computer Science, FOCS 2009
CountryUnited States
CityAtlanta, GA
Period10/25/0910/27/09

ASJC Scopus subject areas

  • Computer Science(all)

Cite this

Khot, S., & Saket, R. (2009). SDP integrality gaps with local ℓ1-embeddability. In Proceedings - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009 (pp. 565-574). [5438596] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS). https://doi.org/10.1109/FOCS.2009.37