The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative-Type Metrics into ℓ ₁

Abstract

In this paper, we disprove a conjecture of Goemans [23] and Linial [36] (also see [6,38]); 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) of Khot [28], 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 (non-uniform) 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 of UNIQUEGAMES 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.

Survey on nonlocal games and operator space theory

“…15, the authors analyze a family of two-player games which shows that the bound provided by the second item in Proposition 4.5 is essentially optimal. The games were originally introduced by Khot and Vishnoi 43 to obtain the first integrality gap between the classical value of a unique game and the value returned by its "basic semidefinite relaxation." The family of games, known as the Khot-Vishnoi games, is parametrized by an integer ℓ and denoted by (KV ℓ ); for each ℓ letting n = 2 ℓ , the game KV ℓ has 2 n /n questions and n answers per player.…”

Superactivation of quantum steering

“…This is due to the fact that both proof methods rely on the nonlocality of an isotropic state, but while Ref. [24] focuses on a specific Bell inequality, namely the Khot-Vishonoi game [22,23], our method considers all steering inequalities for projective measurements via the necessary and sufficient criterion presented in [5].…”

“…Specifically, he proved that certain entangled states ρ admitting a LHV model for general POVMs can be super-activated, in the sense that ρ ⊗k violates a Bell inequality (for some finite k). To derive this result, he took advantage of known Bell inequalities with unbounded quantum violation [21][22][23]. Next, Cavalcanti and colleagues [24] presented a general criterion for k-copy nonlocality.…”

Parameterized Complexity and Approximability of Directed Odd Cycle Transversal

“…We now complement this negative result with a positive approximation result. Recall that due to [23], we know that there is no constant factor approximation for the optimization version of DOCT and hence for Colorful Walk Cover assuming the Unique Games Conjecture. However, we show that if allowed FPT time, then the optimization version of even Colorful Walk Cover can be approximated up to a constant factor.…”

“…On the one hand, Theorem 1 shows that DOCT is intractable from the perspective of parameterized complexity. On the other hand, the problem is known not to admit a constant factor approximation algorithm running in polynomial time, assuming the Unique Games Conjecture [23]. Hence, the next natural question is whether one could get a constant factor approximation algorithm in FPT time.…”