![The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative Type Metrics Into �1 [Extended Abstract]](http://data.docslib.org/img/cb5aedcecbeb557ec5718c0052964610-1.webp)
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. The Leighton- 2We believe that even the uniform version of the ARV-Conjecture is Rao algorithm worked only for the uniform version. false. δ>0 is arbitrary. The integrality gap holds for standard be taken as an argument supporting the UGC). Indeed, the SDPs with triangle inequality constraints. UGC plays a crucial role in our disproval. Let us outline the basic approach we take. First, we build an integrality gap ≈ Theorem 2.4 Let αGW ( 0.878) be the approxima- instance for a natural SDP relaxation of UNIQUE GAMES tion ratio obtained by Goemans-Williamson’s algorithm for (see Figure 5). Surprisingly, we are then able to translate MAXIMUM CUT [17]. For every δ>0, the Goemans- this integrality gap instance into an integrality gap instance Williamson’s SDP has an integrality gap of at-least αGW + of SPARSEST CUT,BALANCED SEPARATOR,MAXIMUM δ, even after including the triangle inequality constraints. CUT and MINIMUM UNCUT. This translation mimics the PCP reduction from the UGC to these problems (note that Ma- This theorem relies on a Fourier analytic result called the same reduction also proves hardness results assuming jority is Stablest Theorem et al. due to Mossel [27]. the UGC)! We believe that this novel approach will have We note that without the triangle inequality constraints, several applications in the future. Already, inspired by αGW + δ Feige and Schechtman [15] already showed an our work, Khot and Naor [21] have proved several non- integrality gap. One more advantage of our result is that embeddability results (e.g. Edit Distance into 1), and Arora it is an explicit construction, where as Feige and Schecht- et al. [2] have constructed integrality gap instances for the man’s construction is randomized (they need to pick ran- MAXQP problem. dom points on the unit sphere). Our result shows that adding the triangle inequality constraints does not add any power to the Goemans-Williamson’s SDP. This nicely complements 2.4. Integrality Gap Instance for the UNIQUE GAMES the result of Khot et al. [20], where it is shown that, as- SDP Relaxation suming the Unique Games Conjecture (UGC), it is NP-hard to approximate MAXIMUM CUT within a factor better than As mentioned above, we construct an integrality gap in- αGW + δ. stance for a natural SDP relaxation of UNIQUE GAMES (see Figure 5). Here, we choose to provide an informal 2.3. Hardness Results for SPARSEST CUT and BAL- description of this construction (the reader should be able ANCED SEPARATOR Assuming the UGC to understand this construction without even looking at the SDP formulation). Our starting point is the hardness of approximation re- sults for cut problems assuming the UGC (see Section C Theorem 2.6 (Informal statement) Let N be an integer and for the statement of the conjecture). We prove the follow- η>0 be a parameter (think of N as large and η as very ing result: tiny). There is a graph G(V,E) of size 2N /N with the fol- lowing properties: Every vertex u ∈ V is assigned a set of Theorem 2.5 Assuming the UGC, SPARSEST CUT and unit vectors B(u):={u1,...,uN } that form an orthonor- BALANCED SEPARATOR (non-uniform versions) are NP- mal basis for the space RN . Further, hard to approximate within any constant factor. 1. For every edge e =(u, v) ∈ E, the set of vectors 3 This particular result was also proved by Chawla et al. B(u) and B(v) are almost the same upto some small INIMUM NCUT [10]. Similar result for M U is implicit in perturbation. To be precise, there is a permutation πe : [19], where the author formulated the UGC and proved → ∀ ≤ ≤ u v ≥ [N] [N], such that 1 i N, πe(i), i the hardness of approximating MIN-2SAT-DELETION.As 1 − η. In other words, for every edge (u, v) ∈ E, mentioned before, Khot et al. [20] proved that the UGC the basis B(u) moves “smoothly/continuously” to the implies αGW + δ hardness result for MAXIMUM CUT.As basis B(v). an aside, we note that the UGC also implies optimal 2 − δ hardness result for VERTEX COVER, as shown in [22]. 2. For any labeling λ : V → [N], i.e., assignment of Therefore, assuming the UGC, all of the above problems an integer λ(u) ∈ [N] to every u ∈ V , for at-least are NP-hard to approximate within respective factors, and − 1 ∈ 1 N η fraction of the edges e =(u, v) E,we hence, the corresponding integrality gap examples must ex- have πe(λ(u)) = λ(v). In other words, no matter how ist (unless P=NP). In particular, if the UGC is true, then we choose to assign a vector uλ(u) ∈ B(u) for every 2 the ( 2, 1,O(1))-Conjecture is false. This is a rather pecu- vertex u ∈ V , the movement from uλ(u) to vλ(v) is liar situation, because the UGC is still unproven, and may “discontinuous” for almost all edges (u, v) ∈ E. very well be false. Nevertheless, we are able to disprove 2 ∪ the ( 2, 1,O(1))-Conjecture unconditionally (which may 3. All vectors in u∈V B(u) have co-ordinates in the set { √1 , √−1 }, and hence, any three of them satisfy the 3We would like to stress that our work was completely independent, N N and no part of our work was influenced by their paper. triangle inequality constraint. The construction is rather non-intuitive: One can walk underlying group is abelian. (But, only in the abelian case, on the graph G by changing the basis B(u) continuously, the Fourier methods work well.) but as soon as one picks a representative vector for each ba- The best known lower bounds for the 2 versus sis, the motion becomes discontinuous almost everywhere! 2 1 question were due to Vempala ( 10 for a metric obtained Of course, one can pick these representatives in a continu- 9 by a computer search), and Goemans (1.024 for a metric ous fashion for any small enough local sub-graph of G,but based on the Leech Lattice), see [29]. Thus, it appeared there is no way to pick representatives in a global fashion. that an entirely new approach was needed to resolve the This construction eventually leads us to a 2 metric which, 2 ( 2, ,O(1))-Conjecture. In this paper, we present an ap- roughly speaking, is locally -embeddable, but globally, 2 1 1 proach based on tools from complexity theory, namely, the it requires super-constant distortion to embed into (such 1 UGC, PCPs, and Fourier analysis of boolean functions. In- local versus global phenomenon has also been observed by terestingly, Fourier analysis is used both to construct the 2 Arora et al. [4]). 2 metric, as well as, to prove the 1 lower bound. 2 3 Difficulty in Proving 2 vs. 1 Lower Bound 2 4. Overview of Our 2 vs. 1 Lower Bound In this section, we describe the difficulties in construct- 2 ing 2 metrics that do not embed well into 1. This might In this section, we present a high level idea of our 2 partly explain why one needs an unusual construction as the 2 versus 1 lower bound (see Theorem 2.1). Given the one in this paper. Our discussion here is informal, without construction of Theorem 2.6, it is fairly straight-forward 2 precise statements or claims. to describe the candidate 2 metric: Let G(V,E) be the RN 2 graph, and B(u) be the orthonormal basis for for ev- Difficulty in constructing 2 metrics: To the best of our ∈ ∈ 2 ery u V , as in Theorem 2.6. Fix s =4. For u V and knowledge, no natural families of metrics are known N 2 x =(x ,...,x ) ∈{−1, 1} , define the vector V x as other than the Hamming metric on {−1, 1}k. The Hamming 1 N u,s, follows: metric is an metric, and hence, not useful for the purposes 1 1 N of obtaining lower bounds. Certain 2 metrics can be V √ u⊗2s 1 2 u,s,x := xi i (1) constructed via Fourier analysis, and one can also construct N i=1 some by solving SDPs explicitly. The former approach has Note that since B(u)={u1,...,uN } is an orthonormal a drawback that metrics obtained via Fourier methods typ- N basis for R , every V x is a unit vector. Fix t to be a ically embed into isometrically. The latter approach has u,s, 1 large odd integer, for instance 2240 +1, and consider the limited scope, since one can only hope to solve SDPsof set of unit vectors S = {V⊗t | u ∈ V, x ∈{−1, 1}N }. moderate size. Feige and Schechtman [15] show that se- u,s,x Using, essentially, the fact that the vectors in ∪ ∈ B(u) are lecting an appropriate number of points from the unit sphere u V a good solution to the SDP relaxation of UNIQUE GAMES, gives a 2 metric. However, in this case, most pairs of points 2 we are able to show that every triple of vectors in S satisfy have distance Ω(1) and hence, the metric is likely to be 1- S 2 the triangle inequality constraint and, hence, defines a 2 embeddable with low distortion. 2 metric. One can also directly show that this 2 metric does 1/6−δ Difficulty in proving 1 lower bounds: To the best of our not embed into 1 with distortion less than (log N) . knowledge, there is no standard technique to prove a lower However, we choose to present our construction in a dif- bound for embedding a metric into 1. The only interesting ferent and a quite indirect way. The (lengthy) presentation (super-constant) lower bound that we know is due to [6, 25], goes through the Unique Games Conjecture, and the PCP where it is shown that the shortest path metric on a constant reduction from UNIQUE GAMES integrality gap instance to degree expander requires Ω(log n) distortion to embed into BALANCED SEPARATOR. Hopefully, our presentation will 1. bring out the intuition as to why and how we came up with 2 General theorems regarding group norms: A group the above set of vectors, which happened to define a 2 met- norm is a distance function d(·, ·) on a group (G, ◦), such ric. At the end, the reader will recognize that the idea of − that d(x, y) depends only on the group difference x ◦ y−1. taking all +/ linear combinations of vectors in B(u) (as Using Fourier methods, it is possible to construct group in Equation (1)) is directly inspired by the PCP reduction. 2 Also, the proof of the 1 lower bound will be hidden inside norms that are 2 metrics. However, it is known that any group norm on Rk, or on any group of characteristic 2, is the soundness analysis of the PCP! isometrically 1-embeddable (see [11]). It is also known The overall construction can be divided into three steps: (among the experts in this area) that such a result holds for every abelian group. Therefore, any approach, just via 1. A PCP reduction from UNIQUE GAMES to BAL- group norms would be unlikely to succeed, as long as the ANCED SEPARATOR. 2. Constructing an integrality gap instance for a natural The integrality gap is defined to be α/γ. (Thus, we desire SDP relaxation of UNIQUE GAMES. that γ α.)
3. Combining these two to construct an integrality gap in- The next three sections describe the three steps involved stance of BALANCED SEPARATOR. This also gives a in constructing an integrality gap instance of BALANCED 2 1/6−δ 2 metric that needs (log log n) distortion to em- SEPARATOR. Once that is done, it follows from a folk- bed into 1. 2 lore result that the resulting 2 metric (defined by vectors {v | i ∈ V }) requires distortion at-least Ω(α/γ) to embed We present an overview of each of these steps in three i into . This would prove Theorem 2.1 with an appropriate separate sections. Before we do that, let us summarize the 1 choice of parameters. precise notion of an integrality gap instance of BALANCED SEPARATOR. To keep things simple in this exposition, we will pretend as if our construction works for the uniform 4.2. The PCP Reduction from UNIQUE GAMES to version of BALANCED SEPARATOR as well. (Actually, it BALANCED SEPARATOR doesn’t. We have to work with the non-uniform version and it complicates things a little.) An instance U(G(V,E), [N], {πe}e∈E) of UNIQUE GAMES consists of a graph G(V,E) and permutations 4.1. SDP Relaxation of BALANCED SEPARATOR πe :[N] → [N] for every edge e =(u, v) ∈ E.The goal is to find a labeling λ : V → [N] that satisfies as Given a graph G (V ,E ),BALANCED SEPARATOR many edges as possible. An edge e =(u, v) is satisfied if 1 1 πe(λ(u)) = λ(v).LetOPT(U) denote the maximum frac- asks for a ( 2 , 2 )-partition of V that cuts as few edges as possible. (However, the algorithm is allowed to output a tion of edges satisfied by any labeling. roughly balanced partition, say ( 1 , 3 )-partition.) Following 4 4 UGC (Informal Statement): It is NP-hard to decide is an SDP relaxation of BALANCED SEPARATOR: whether an instance U of UNIQUE GAMES has OPT(U) ≥ 1−η (YES instance) or OPT(U) ≤ ζ (NO instance), where 1 1 2 Minimize vi − vj (2) η,ζ > 0 can be made arbitrarily small by choosing N to be |E| 4 e ={i,j}∈E a sufficiently large constant. Subject to It is possible to construct an instance of BAL- 2 ANCED SEPARATOR Gε(V ,E ) from an instance ∀ i ∈ V vi =1 (3) U(G(V,E), [N], {πe}e∈E) of UNIQUE GAMES.We ∀ i, j, l ∈ V v − v 2 + v − v 2 ≥v − v 2(4) i j j l i l describe only the high level idea here. The construction is v − v 2 ≥| |2 N i
Note that a {+1, −1}-valued solution represents a true We let V := ∪u∈V V [u]. For every edge e =(u, v) ∈ E, partition, and hence, this is an SDP relaxation. Constraint the graph Gε has edges between the blocks V [u] and V [v]. (4) is the triangle inequality constraint and Constraint (5) These edges are supposed to capture the constraint that the stipulates that the partition be balanced. The notion of inte- labels of u and v are consistent (i.e. πe(λ(u)) = λ(v)). grality gap is summarized in the following definition: Roughly speaking, a vertex (u, x) ∈ V [u] is connected to avertex(v,y) ∈ V [v] if and only if, after identifying the Definition 4.1 An integrality gap instance of BALANCED co-ordinates in [N] via the permutation πe, the Hamming SEPARATOR is a graph G (V ,E ) and an assignment of distance between the bit-strings x and y is at-most εN. unit vectors i → vi to its vertices such that: This reduction has the following two properties: • 1 3 Every almost balanced partition (say ( 4 , 4 )-partition; the choice is arbitrary) of V cuts at-least α fraction of Theorem 4.2 (PCP reduction: Informal statement) edges. 1. (Completeness/YES case): If OPT(U) ≥ 1 − η, then • {v | ∈ } 1 1 The set of vectors i i V satisfy (3)-(5), and the the graph Gε has a ( 2 , 2 )-partition that cuts at-most SDP objective value in Equation (2) is at-most γ. η + ε fraction of its edges. −O(1/ε2) N 2. (Soundness/NO Case): If OPT(U) ≤ 2 √ , then of G is (u, x), where u ∈ V and x ∈{−1, 1} .Tothis 1 3 V⊗t every ( 4 , 4 )-partition of Gε cuts at-least ε fraction vertex, we attach the unit vector u,s,x (for s =4,t = of its edges. 2240 +1), where
Remark 4.3 We were imprecise on two counts: (1) The N V √1 u⊗2s soundness property holds only for those partitions that par- u,s,x := xi i . N tition a constant fraction of the blocks V [u] in a roughly i=1 balanced way. We call such partitions piecewise balanced. It can be shown that the set of vectors {V⊗t | u ∈ This is where the issue of uniform versus non-uniform ver- u,s,x V, x ∈{−1, 1}N } satisfy the triangle inequality constraint, sion of BALANCED SEPARATOR arises. (2) For the sound- 2 V⊗t V⊗t ness property, we can only claim that every piecewise bal- and hence, defines a 2 metric. Vectors u,s,x and u,s,−x t are antipodes of each other, and hence, the SDP Constraint anced partition cuts at least ε fraction of edges, where√ any 1 (5) is also satisfied. Finally, we show that the SDP objective t> 2 can be chosen in advance. Instead, we write ε for the simplicity of notation. value (Expression (2)) is O(η + ε). It suffices to show that for every edge ((u, x), (v,y)) in G(V ,E),wehave 4.3. Integrality Gap Instance for the UNIQUE GAMES ⊗t ⊗t V x, V y ≥ 1 − O(st(η + ε)). SDP Relaxation u,s, v,s, This holds because, whenever ((u, x), (v,y)) is an edge of This has already been described in Theorem 2.6. The G, we have (after identifying the indices via the permuta- graph G(V,E) therein along with the ortho-normal basis → u v ≥ − tion πe :[N] [N]): (a) πe(i), i 1 η, for all B(u), for every u ∈ V, can be used to construct an instance 1 ≤ i ≤ N. (b) The Hamming distance between x and y is U(G(V,E), [N], {πe}e∈E) of UNIQUE GAMES. For every at-most εN. edge e =(u, v) ∈ E, we have an (unambiguously defined) → u v ≥ − permutation πe :[N] [N], where πe(i), i 1 η, 4.5. Quantitative Parameters for all 1 ≤ i ≤ N. OPT U ≤ 1 Theorem 2.6 implies that ( ) N η .Onthe It follows from above discussion (see also Definition other hand, the fact that for every edge e =(u, v), 4.1) that the integrality gap for BALANCED SEPARATOR is √ 2 the bases B(u) and B(v) are very close to each other Ω(1/ ε) provided that η ≈ ε, and N η > 2O(1/ε ). We can U − means that the SDP objective value for is at-least 1 choose η ≈ ε ≈ (log N)−1/3. Since the size of the graph η. SDP (Formally, the objective value is defined to be G is at-most n =22N , we see that the integrality gap is 1 N u v 1/6 Ee=(u,v)∈E N i=1 πe(i), i .) ≈ (log log n) as desired. Thus, we have a concrete instance of UNIQUE GAMES 1 with optimum at most N η = o(1), and which has an SDP 4.6. Proving the Triangle Inequality solution with objective value at-least 1 − η. This is what an integrality gap example means: The SDP solution cheats in As mentioned above, one can show that the set of vec- {V⊗t | ∈ x ∈{− }N } an unfair way! tors u,s,x u V, 1, 1 satisfy the trian- gle inequality constraints. This is the most technical part 4.4. Integrality Gap Instance for the BALANCED SEP- of the paper, but we would like to stress that this is where ARATOR SDP Relaxation the “magic” happens. In our construction, all vectors in N ∪u∈V B(u) happen to be points of√ the hypercube {−1, 1} Now we combine the two modules described above. We (upto a normalizing factor of 1/ N), and therefore, they U { } take the instance (G(V,E), [N], πe e∈E) as above, and define an 1 metric. The apparently outlandish operation run the PCP reduction on it. This gives us an instance of taking their +/− combinations combined with tensor- 2 G (V ,E ) of BALANCED SEPARATOR. We show that this ing, miraculously leads to a metric that is ( 2 and) non- 1- is an integrality gap instance in the sense of Defintion 4.1. embeddable. Since U is a NO instance of UNIQUE GAMES (i.e. OPT(U)=o(1)), Theorem 4.2 implies that every√ (piece- 5. Acknowledgments wise) balanced partition of G must cut at-least ε fraction η −O(1/ε2) of the edges. We need to have 1/N ≤ 2 for this We are very thankful to Assaf Naor and James Lee for to hold. useful pointers and for ruling out some of our initial ap- On the other hand, we can construct an SDP solution for proaches. Many thanks to Sanjeev Arora, Moses Charikar, the BALANCED SEPARATOR instance which has an objec- Umesh Vazirani, Ryan O’Donnell and Elchanan Mossel for tive value of at-most O(η + ε). Note that a typical vertex insightful discussions at various junctures. References [17] M. X. Goemans and D. P. Williamson. Improved approxi- mation algorithms for maximum cut and satisfiability prob- lems using semidefinite programming. 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Random d(y,x). (4) For all x, y, z X, d(x, y)+d(y,z) d(x, z). Struct. Algorithms, 20(3):403–440, 2002. (X, d) is said to be a finite metric space if X is finite. (X, d) [16] M. X. Goemans. Semidefinite programming in combinato- is called a semi-metric space if one allows d(x, y)=0even rial optimization. Math. Program., 79:143–161, 1997. when x = y. Definition A.2 (X1,d1) embeds with distortion at-most Γ Remark A.9 The versions of SPARSEST CUT and BAL- into (X2,d2) if there exists a map φ : X1 → X2 such that ANCED SEPARATOR that we define below are non-uniform for all x, y ∈ Xd1(x, y) ≤ d2(φ(x),φ(y)) ≤ Γ · d1(x, y). versions with demands. The uniform version has all de- If Γ=1, then (X1,d1) is said to isometrically embed in mands equal to 1 (i.e. unit demand for every pair of ver- (X2,d2). tices). The metrics we would be concerned with are: (1) metrics:ForX ⊆ Rm, for some m ≥ 1, and x, y ∈ The Sparsest Cut Problem p X, (x, y)=( m |x − y |p)1/p . Here, p ≥ 1, and the p i=1 i i Definition A.10 (SPARSEST CUT) For a graph G = x y m | − | metric ∞( , ) = maxi=1 xi yi . (V,E) with a weight wt(e), and a demand dem(e) as- (2) Cut (semi-)metrics: A cut metric δS on a set X, defined sociated to each edge e ∈ E, the goal is to optimize by the set S ⊆ X is: wt e∈E(S,S) (e) min∅= SV dem(e) . 1if|{x, y}∩S| =1 e∈E(S,S) δS(x, y)= 0 otherwise It follows from Fact A.4 that the objective function above is wt e={x,y}∈E (e)d(x,y) the same as mind∈1 . Denote this The cut-cone (denoted CUTn) is the cone generated by e={x,y}∈E dem(e)d(x,y) CUT cut metrics on an n-point set X. Formally, n := minimum for {G, wt, dem} by ψ1(G). Consider the fol- { ≥ ⊆ } S λSδS : λS 0 for all S X . To avoid referring to lowing two quantities associated to {G, wt, dem}: the dimension, denote CUT:= ∪nCUTn. wt (3) Negative type metrics:√ A metric space (X, d) is said to e={x,y}∈E (e)d(x, y) ψ∞(G) := min , be of negative type if (X, d) embeds isometrically into 2. d∈∞ { }∈ dem(e)d(x, y) v ∈ Rm e= x,y E Formally, there is an integer m and a vector x for ∈ v − v 2 every x X, such that d(x, y)= x y . The class of { }∈ wt(e)d(x, y) 2 e= x,y E all negative type metrics is denoted by . ψneg(G) := min . 2 ∈ 2 dem d 2 e={x,y}∈E (e)d(x, y)
A.2. Facts about Metric Spaces Facts A.5 and A.3 imply that ψ1(G) ≥ ψneg(G) ≥ ψ∞(G). In addition, Bourgain’s Embedding Theorem (Theorem Fact A.3 [11] Any finite metric space isometrically embeds A.6) can be used to show that ψ1(G) ≤ O(log n) · ψ∞(G), into ∞. where n := |V |. Fact A.7 implies that this factor of O(log n) is tight upto a constant. Fact A.4 [11] (X, d) is 1 embeddable iff d ∈ CUT. It is also the case that ψneg(G) is efficiently computable ⊆ 2 Fact A.5 [11] 1 2. using a semi-definite program (SDP) of Figure 2. Let the { } Theorem A.6 (Bourgain’s Embedding Theorem [7]) vertex set be V = 1, 2,...,n . For a metric d on V, let Q := Q(d) be the matrix whose (i, j)-th entry is Q[i, j]:= Any n-point metric space embeds into 1 with distortion 1 (d(i, n)+d(j, n) − d(i, j)) . at-most Cb log n, for some absolute constant Cb. 2 Fact A.7 [6, 25] There is an n-point metric, any embed- Minimize wt(e)d(i, j) ding of which into 1, requires Ω(log n) distortion. e={i,j}
2 A.3. The ( 2, 1,O(1))-Conjecture Subject to d(·, ·) is a metric 2 dem Conjecture A.8 (( 2, 1,O(1))-Conjecture, [16, 24]) e={i,j} (e)d(i, j)=1 Every 2 metric can be embedded into with distortion 2 1 Q(d) is positive semidefinite at-most Cneg, for some absolute constant Cneg ≥ 1.
A.4. Cut Problems and their SDP Relaxations Figure 2. SDP relaxation of SPARSEST CUT
In this section, we define the cut problems that we study 2 and present their SDP relaxations. All graphs are com- Fact A.11 Suppose that every n-point 2 metric embeds plete undirected graphs and may have multiple edges or into 1 with distortion f(n). Then, ψ1(G) ≤ f(n) · self loops, unless mentioned otherwise, with non-negative ψneg(G), and SPARSEST CUT can be approximated to 2 weights or demands associated to its edges. For a graph within a factor of f(n). In particular, if the ( 2, 1,O(1))- G =(V,E), and S ⊆ V ,letE(S, S) denote the set of Conjecture (Conjecture A.8) is true, then there is a constant edges with one endpoint in S and other in S. factor approximation algorithm for SPARSEST CUT. 1 The Balanced Separator Problem Maximize wt(e)v − v 2 4 x y Definition A.12 (BALANCED SEPARATOR) For a graph e={x,y} wt dem G =(V,E) with a weight (e), and a demand (e) Subject to ∈ dem associated to each edge e E, let D := e∈E (e) 2 be the total demand. Let a balance parameter B be given ∀x ∈ V vx =1 ≤ ≤ where D/4 B D/2. The goal is to find a non- ∀ x, y, z ∈ V v − v 2 + v − v 2 ≥v − v 2 wt x y y z x z trivial cut (S, S) that minimizes e∈E(S,S) (e), sub- dem ≥ ject to e∈E(S,S) (e) B. The cuts that satisfy dem ≥ Figure 4. SDP relaxation of MAXIMUM CUT e∈E(S,S) (e) B are called B-balanced cuts. Figure 3 is an SDP relaxation of BALANCED SEPARATOR with parameter B. Goemans and Williamson [17] gave an αGW (≈ 0.878) ap- proximation algorithm for MAXIMUM CUT. They showed 1 that every SDP solution with objective value γSDP can be Minimize wt(e)v − v 2 4 x y rounded to a cut in the graph that cuts edges with weight e={x,y} ≥ αGW · γSDP. We note here that their rounding procedure Subject to does not make use of the triangle inequality constraints.
2 ∀x ∈ V vx =1 The Minimum Uncut Problem ∀ x, y, z ∈ V v − v 2 + v − v 2 ≥v − v 2 x y y z x z Definition A.18 (MINIMUM UNCUT) Given a graph G = 1 2 wt ∈ { } dem(e)vx − vy ≥ B (V,E) with a weight (e) associated to each edge e E, 4 e= x,y wt e∈E (S,S)∪E(S,S) (e) the goal is to optimize min∅ . =S V e∈E wt(e)
Figure 3. SDP relaxation of BALANCED SEPA- The semi-definite relaxation of MINIMUM UNCUT is simi- RATOR lar to that of MAXIMUM CUT except the objective function in Figure 4 is replaced by We need the following (folk-lore) result stating that, start- 1 ing with a good SDP solution to Figure 3, one can find a Minimize 1 − wt(e)v − v 2 4 x y balanced partition in a graph by iteratively finding (approx- e={x,y} imate) sparsest cut in the graph. Goemans and Williamson [17] showed that every SDP so- Theorem A.13 Suppose x → v is a solution for the SDP x lution for MINIMUM UNCUT with objective value βSDP can of Figure 3 with objective value 1 wt(e)v − 4 e={x,y} x be rounded to a cut in the graph,√ such that the weight of v 2 ≤ 2 y ε. Assume that the 2 metric defined by the vectors edges left uncut is at-most O( βSDP). We note again that {v | ∈ } x x V embeds into 1 with distortion f(n) (n = their rounding procedure does not make use of the triangle |V |). Then, there exists a B -balanced cut (S, S), B ≥ inequality constraints. wt ≤ · B/3 such that e∈E(S,S) (e) O(f(n) ε). B. Overall Strategy for Disproving the The ARV-Conjecture 2 ( 2, 1,O(1))-Conjecture Conjecture A.14 (Uniform Version) The integrality gap of the SDP of Figure 1 is O(1). We describe the high-level approach to our disproval of the ( 2, ,O(1))-Conjecture in this section. We construct Conjecture A.15 (Non-Uniform Version) The integrality 2 1 an integrality gap instance of non-uniform BALANCED gap of the SDP of Figure 3 is O(1). SEPARATOR to disprove the non-uniform ARV-Conjecture, 2 and that suffices to disprove the ( 2, ,O(1))-Conjecture Fact A.16 The ( 2, 1,O(1))-Conjecture implies the non- 2 1 uniform ARV-Conjecture (Conjecture A.15). using the (folk-lore) Fact A.16. We construct a complete weighted graph G(V,wt), The Maximum Cut Problem with vertex set V and weight wt(e) on edge e, and with wt e (e)=1. The vertex set is partitioned into sets Definition A.17 (MAXIMUM CUT) For a graph G = V ,V ,...,V , each of size |V |/r (think of r ≈ |V |). wt ∈ 1 2 r (V,E) with a weight (e) associated to each edge e E, → wt A cut A in the graph is viewed as a function A : V e∈E(S,S) (e) the goal is to optimize max∅ . {− } =S V e∈E wt(e) 1, 1 . We are interested in cuts that cut many sets Vi in a somewhat balanced way: For 0 ≤ θ ≤ 1, a cut A is called C. The Unique Games Conjecture (UGC) E E ≤ θ-piecewise balanced if i∈R[r] x∈RVi [A(x)] θ. Definition C.1 (UNIQUE GAMES) An instance We also assign a unit vector to every vertex in the graph. Let U (G(V,E), [N], {π } ∈ , wt) of UNIQUE GAMES v denote the vector assigned to vertex x. Our construction e e E x is defined as follows: G =(V,E) is a a graph with of the graph G(V,wt) and the vector assignment x → v x a set of vertices V and a set of edges E, with possibly can be summarized as follows: parallel edges. An edge e whose endpoints are v and w is written as e{v,w}. For every e ∈ E, there is a bijection → wt ∈ R+ Theorem B.1 Fix any 1