Abstract

In this paper we disprove the following conjecture due to Goemans [16] and Linial [24] (also see [5, 26]): “Every negative type metric embeds into ℓ1 with constant distortion.” We show that for every δ>0, and for 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) of Khot [19], establishing a previously unsuspected connection between PCPs and the theory of metric embeddings. We first prove that the UGC implies super-constant hardness results for (non-uniform) SPARSEST CUT and MINIMUM UNCUT problems. It is already known that the UGC also implies an optimal hardness result for MAXIMUM CUT [20]. Though these hardness results depend on the UGC, the integrality gap instances rely “only ” on the PCP reductions for the respective problems. Towards this, we first construct an integrality gap instance for a natural SDP relaxation of UNIQUE GAMES. Then, we “simulate ” the PCP reduction and “translate ” the integrality gap instance of UNIQUE GAMES to integrality gap instances for the respective cut problems! This enables us to prove a (log log n) 1/6−δ integrality gap for (non-uniform) SPARSEST CUT and MIN-IMUM UNCUT, and an optimal integrality gap for MAX-IMUM CUT. All our SDP solutions satisfy the so-called “triangle inequality ” constraints. This also shows, for the first time, that the triangle inequality constraints do not add any power to the Goemans-Williamson’s SDP relaxation of MAXIMUM CUT. The integrality gap for SPARSEST CUT immediately implies a lower bound for embedding negative type metrics into ℓ1. It also disproves the non-uniform version of Arora, Rao and Vazirani’s Conjecture [5], asserting that the integrality gap of the SPARSEST CUT SDP, with the triangle inequality constraints, is bounded from above by a constant.

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