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

• Authors: Subhash A. Khot;Nisheeth K. Vishnoi
• Venue: FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
• Year: 2005

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 \ell 1 with constant distortion." We show that for every \delta} \ge 0, and for large enough n, there is an n-point negative type metric which requires distortion at-least \log \log n^{1/6 - \delta} to embed into \ell 1. 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 - \delta} integrality gap for (non-uniform) SPARSEST CUT and MINIMUM UNCUT, and an optimal integrality gap for MAXIMUM 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驴s SDP relaxation of MAXIMUM CUT. The integrality gap for SPARSEST CUT immediately implies a lower bound for embedding negative type metrics into \ell 1. It also disproves the non-uniform version of Arora, Rao and Vazirani驴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.