Integrality gaps for sparsest cut and minimum linear arrangement problems - Devanur, Nikhil R.; Khot, Subhash; Saket, Rishi; Vishnoi, Nisheeth; Devanur, Nikhil (STOC, 2006)
Arora, Rao and Vazirani showed that the standard semi-definite programming (SDP) relaxation of the sparsest cut problem with the triangle inequality constraints has an integrality gap of O([n]). They conjectured that the gap is bounded from above by a constant. In this paper, we disprove this conjecture by constructing an Ω(n) integrality gap instance. Khot and Vishnoi had earlier disproved the non-uniform version of this Conjecture. A simple ``stretching'' of the integrality gap instance for the sparsest cut problem serves as an Ω(n) integrality gap instance for the SDP relaxation of the Minimum Linear Arrangement problem. This SDP relaxation was considered in Charikar et. al. and Feige and Lee, where it was shown that its integrality gap is bounded from above by O([n] n).