Integrality gaps for sparsest cut and minimum linear arrangement problems

Abstract

Arora, Rao and Vazirani [2] showed that the standard semidefinite programming (SDP) relaxation of the Sparsest Cut problem with the triangle inequality constraints has an integrality gap of O( √ log n). They conjectured that the gap is bounded from above by a constant. In this paper, we disprove this conjecture (referred to as the ARV-Conjecture) by constructing an Ω(log log n) integrality gap instance. Khot and Vishnoi [16] had earlier disproved the non-uniform version of the ARV-Conjecture.A simple "stretching" of the integrality gap instance for the Sparsest Cut problem serves as an Ω(log log n) integrality gap instance for the SDP relaxation of the Minimum Linear Arrangement problem. This SDP relaxation was considered in [6,11], where it was shown that its integrality gap is bounded from above by O( √ log n log log n).

On Khot’s unique games conjecture

“…This has been a rather unique approach to the construction of counterexamples in metric geometry. The lower bound was improved to Ω(log log n) by Krauthgamer and Rabani [27], and shortly afterward Devanur, Khot, Saket, and Vishnoi [18] showed that even the Sparsest Cut relaxation has an integrality gap Ω(log log n).…”

Inapproximability Results for Sparsest Cut, Optimal Linear Arrangement, and Precedence Constrained Scheduling

“…This improves over the O(log n)-approximation of Rao and Richa. These SDP relaxations were shown to have integrality gap Ω(log log n) by Devanur, Khot, Saket & Vishnoi [10]. As noted in [10], it remains a challenging open problem to prove a hardness of approximation result for Sparsest Cut and OLA.…”

“…These SDP relaxations were shown to have integrality gap Ω(log log n) by Devanur, Khot, Saket & Vishnoi [10]. As noted in [10], it remains a challenging open problem to prove a hardness of approximation result for Sparsest Cut and OLA. The currently known hardness results apply only 1 Also known as Minimum Linear Arrangement.…”

On the Unique Games Conjecture

“…These connections are further explained in Section 8.2. The KKL Theorem has been used to prove a super-constant inapproximability result for the non-uniform Sparsest Cut problem by Chawla et al [25] and to construct Ω(log log N ) integrality gaps for a natural SDP relaxation for the non-uniform as well as the uniform Sparsest Cut problem [75,33], improving on the integrality gap obtained earlier in [71].…”