The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative Type Metrics into 1 [Extended Abstract] Subhash A. Khot Nisheeth K. Vishnoi College of Computing, Georgia Tech IBM India Research Lab Atlanta GA 30332 Block-1 IIT Delhi, New Delhi 110016 [email protected] [email protected] Abstract In this paper we disprove the following conjecture due to Goemans [16] and Linial [24] (also see [5, 26]): “Every 1. Introduction negative type metric embeds into 1 with constant distor- tion.” We show that for every δ>0, and for large enough In recent years, the theory of metric embeddings has n, there is an n-point negative type metric which requires 1/6−δ played an increasing role in algorithm design. Best approx- distortion at-least (log log n) to embed into 1. imation algorithms for several NP-hard problems rely on Surprisingly, our construction is inspired by the Unique techniques (and theorems) used to embed one metric space Games Conjecture (UGC) of Khot [19], establishing a pre- into another with low distortion. viously unsuspected connection between PCPs and the the- Bourgain [7] showed that every n-point metric embeds ory of metric embeddings. We first prove that the UGC into 1 (in fact into 2) with distortion O(log n). Indepen- implies super-constant hardness results for (non-uniform) dently, Aumann and Rabani [6] and Linial, London and Ra- SPARSEST CUT and MINIMUM UNCUT problems. It is al- binovich [25] gave a striking application of Bourgain’s The- ready known that the UGC also implies an optimal hard- orem: An O(log n) approximation algorithm for SPARS- ness result for MAXIMUM CUT [20]. EST CUT. The approximation ratio is exactly the distor- Though these hardness results depend on the UGC,the tion incurred in Bourgain’s Theorem. This gave an alter- integrality gap instances rely “only” on the PCP reductions nate approach to the seminal work of Leighton and Rao for the respective problems. Towards this, we first construct [23], who obtained an O(log n) approximation algorithm an integrality gap instance for a natural SDP relaxation for SPARSEST CUT via a LP-relaxation based on muticom- of UNIQUE GAMES. Then, we “simulate” the PCP reduc- modity flows. It is well-known that an f(n) factor algo- tion and “translate” the integrality gap instance of UNIQUE rithm for SPARSEST CUT can be used iteratively to de- GAMES to integrality gap instances for the respective cut sign an O(f(n)) factor algorithm for BALANCED SEPA- problems! This enables us to prove a (log log n)1/6−δ in- 1 1 RATOR: Given a graph that has a ( 2 , 2 )-partition cutting an tegrality gap for (non-uniform) SPARSEST CUT and MIN- 1 2 α fraction of the edges, the algorithm produces a ( 3 , 3 )- IMUM UNCUT, and an optimal integrality gap for MAX- partition that cuts at-most O(f(n)α) fraction of the edges. IMUM CUT. All our SDP solutions satisfy the so-called Such partitioning algorithms are very useful as sub-routines “triangle inequality” constraints. This also shows, for the in designing graph theoretic algorithms via the divide-and- first time, that the triangle inequality constraints do not add conquer paradigm. any power to the Goemans-Williamson’s SDP relaxation of The results of [6, 25] are based on the metric LP relax- MAXIMUM CUT. ation of SPARSEST CUT. Given an instance G(V,E) of The integrality gap for SPARSEST CUT immediately im- SPARSEST CUT,letdG be the n-point metric obtained as a plies a lower bound for embedding negative type metrics solution to this LP. The metric dG is then embedded into 1 into 1. It also disproves the non-uniform version of Arora, via Bourgain’s Theorem. Since 1 metrics are non-negative Rao and Vazirani’s Conjecture [5], asserting that the in- linear combinations of cut metrics, an embedding into 1 tegrality gap of the SPARSEST CUT SDP, with the triangle essentially gives the desired sparse cut (up to an O(log n) inequality constraints, is bounded from above by a constant. approximation factor). Subsequent to this result, it was real- ized that one could write an SDP relaxation of SPARSEST Conjecture. The main result in this paper is a disproval of 2 CUT, and enforce an additional condition, that the metric the (2,1,O(1))-Conjecture and a disproval of the non- dG belong to a special class of metrics, called the negative uniform version of the ARV-Conjecture, see Conjecture 2 2 2 type metrics (denoted by 2). Clearly, if 2 embeds into 1 A.15. The disprovals follow from the construction of a with distortion g(n), then one would get a g(n) approxima- super-constant integrality gap for the non-uniform version tion to SPARSEST CUT.1 of BALANCED SEPARATOR (which implies the same gap 2 The results of [6, 25] led to the conjecture that 2 em- for the non-uniform version of SPARSEST CUT). We also beds into 1 with distortion Cneg, for some absolute con- obtain integrality gap instances for MAXIMUM CUT and stant Cneg. This conjecture has been attributed to Goemans MINIMUM UNCUT. In the following sections, we describe 2 [16] and Linial [24], see [5, 26]. This conjecture, which we our results in detail and present an overview of our 2 versus 2 will henceforth refer to as the (2,1,O(1))-Conjecture, if 1 lower bound. Due to the lack of space, the only techni- true, would have had tremendous algorithmic applications cal details we are able to provide in this version of the pa- (apart from being an important mathematical result). Sev- per are: The basic setup, which relates cuts and metrics, in eral problems, specifically cut problems (see [11]), can be Appendix A, and the overall approach for the disproval of 2 formulated as optimization problems over the class of 1 (2,1,O(1))-Conjecture in Appendix B. The full version metrics, and optimization over 1 is an NP-hard problem of this paper will have all the details. 2 in general. However, one can optimize over 2 metrics in ⊆ 2 2 polynomial time via SDPs (and 1 2). Hence, if 2 was 2. Our Results embeddable into 1 with constant distortion, one would get a computationally efficient approximation to 1 metrics. 2 2 2.1. The Disproval of (2,1,O(1))-Conjecture However, no better embedding of 2 into 1, other than Bourgain’s O(log n) embedding (that works for all met- We prove the following theorem which follows from the rics), was known until recently. A breakthrough result of √ integrality gap construction for non-uniform BALANCED Arora, Rao and Vazirani (ARV) [5] gave an O( log n) ap- SEPARATOR. See Section A for definitions and basic facts. proximation to (uniform) SPARSEST CUT by showing√ that the integrality gap of the SDP relaxation is O( log n) Theorem 2.1 For every δ>0 and for all sufficiently large (see also [28] for an alternate perspective on ARV). Sub- 2 n, there is an n-point 2 metric which cannot be embedded sequently, ARV techniques were used by Chawla, Gupta 1/6−δ into 1 with distortion less than (log log n) . and Racke¨ [9] to give an O(log3/4 n) distortion embed- 2 ding of 2 metrics into 2, and√ hence, into 1. This re- Remark 2.2 One of the crucial ingredients for obtaining sult was further improved to O( log n log log n) by Arora, the lower bound of (log log n)1/6−δ in Theorems 2.1 and Lee, and Naor [3]. The latter paper implies, in particu- 2.3 is Bourgain’s Junta Theorem [8]. A recent improvement lar, that every√ n-point 1 metric embeds into 2 with dis- of this theorem due to Mossel et al. [27] improves both of tortion√ O( log n log log n), almost matching decades old our lower bounds to (log log n)1/4−δ. Ω( log n) lower bound due to Enflo [12]. Techniques√ from ARV have also been applied, to obtain O( log n) approx- 2.2. Integrality Gap Instances for Cut Problems imation to MINIMUM UNCUT and related problems [1], to VERTEX SEPARATOR [13], and to obtain a 2 − O( √ 1 ) log n SPARSEST CUT and BALANCED SEPARATOR (non- approximation to VERTEX COVER [18]. It was conjec- uniform versions), as well as MAXIMUM CUT and MINI- tured in the ARV paper, that the integrality gap of the SDP MUM UNCUT are defined in Section A.4. Natural SDP re- relaxation of SPARSEST CUT is bounded from above by laxations for these problems are also described there. All an absolute constant (they make this conjecture only for the SDPs include the so-called triangle inequality con- 2 the uniform version, and the (2,1,O(1))-Conjecture im- straints: For every triple of vectors u, v, w in the SDP so- plies it also for the non-uniform version). Thus, if the lution, u − v2 + v − w2 ≥u − w2. Note that 2 (2,1,O(1))-Conjecture and/or the ARV-Conjecture were these constraints are always satisfied by the integral solu- true, one would potentially get a constant factor approxi- tions, i.e., +1, −1 valued solutions. We prove the following mation to a host of problems, and perhaps, an algorithm two theorems: for VERTEX COVER with an approximation factor better than 2! Clearly, it is an important open problem to prove Theorem 2.3 SPARSEST CUT, BALANCED SEPARATOR 2 or disprove the (2,1,O(1))-Conjecture and/or the ARV- (non-uniform versions of both) and MINIMUM UNCUT have an integrality gap of at-least (log log n)1/6−δ, where 1Algorithms based on metric embeddings (typically) work for the non- uniform version of SPARSEST CUT, which is more general.