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53
Optimal inapproximability results for MAXCUT and other 2variable CSPs?
, 2005
"... In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games ..."
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Cited by 238 (32 self)
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In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games
The Unique Games Conjecture, integrality gap for cut problems and embeddability of negative type metrics into `1
 In Proc. 46th IEEE Symp. on Foundations of Comp. Sci
, 2005
"... In this paper we disprove the following conjecture due to Goemans [17] and Linial [25] (also see [5, 27]): “Every negative type metric embeds into `1 with constant distortion. ” We show that for every δ> 0, and for large enough n, there is an npoint negative type metric which requires distortion ..."
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Cited by 180 (13 self)
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In this paper we disprove the following conjecture due to Goemans [17] and Linial [25] (also see [5, 27]): “Every negative type metric embeds into `1 with constant distortion. ” We show that for every δ> 0, and for large enough n, there is an npoint negative type metric which requires distortion atleast (log log n)1/6−δ to embed into `1. Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC) of Khot [20], establishing a previously unsuspected connection between PCPs and the theory of metric embeddings. We first prove that the UGC implies superconstant hardness results for (nonuniform) Sparsest Cut and Minimum Uncut problems. It is already known that the UGC also implies an optimal hardness result for Maximum Cut [21]. Though these hardness results rely on the UGC, we demonstrate, nevertheless, that the corresponding PCP reductions can be used to construct “integrality gap instances ” 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
Nonembeddability theorems via Fourier analysis
"... Various new nonembeddability results (mainly into L1) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on {0, 1}d has L1 distortion (log d) 12o(1). We also give new lower bounds on the L1 distortion of flat tori, quotients of the discrete hypercube under group ac ..."
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Cited by 53 (12 self)
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Various new nonembeddability results (mainly into L1) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on {0, 1}d has L1 distortion (log d) 12o(1). We also give new lower bounds on the L1 distortion of flat tori, quotients of the discrete hypercube under group actions, and the transportation cost (Earthmover) metric.
Conditional hardness for approximate coloring
 In STOC 2006
, 2006
"... We study the APPROXIMATECOLORING(q, Q) problem: Given a graph G, decide whether χ(G) ≤ q or χ(G) ≥ Q (where χ(G) is the chromatic number of G). We derive conditional hardness for this problem for any constant 3 ≤ q < Q. For q ≥ 4, our result is based on Khot’s 2to1 conjecture [Khot’02]. For ..."
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Cited by 46 (13 self)
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We study the APPROXIMATECOLORING(q, Q) problem: Given a graph G, decide whether χ(G) ≤ q or χ(G) ≥ Q (where χ(G) is the chromatic number of G). We derive conditional hardness for this problem for any constant 3 ≤ q < Q. For q ≥ 4, our result is based on Khot’s 2to1 conjecture [Khot’02]. For q = 3, we base our hardness result on a certain ‘⊲< shaped ’ variant of his conjecture. We also prove that the problem ALMOST3COLORINGε is hard for any constant ε> 0, assuming Khot’s Unique Games conjecture. This is the problem of deciding for a given graph, between the case where one can 3color all but a ε fraction of the vertices without monochromatic edges, and the case where the graph contains no independent set of relative size at least ε. Our result is based on bounding various generalized noisestability quantities using the invariance principle of Mossel et al [MOO’05].
Integrality gaps for strong SDP relaxations of unique games
"... Abstract — With the work of Khot and Vishnoi [18] as a starting point, we obtain integrality gaps for certain strong SDP relaxations of Unique Games. Specifically, we exhibit a Unique Games gap instance for the basic semidefinite program strengthened by all valid linear inequalities on the inner pro ..."
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Cited by 42 (8 self)
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Abstract — With the work of Khot and Vishnoi [18] as a starting point, we obtain integrality gaps for certain strong SDP relaxations of Unique Games. Specifically, we exhibit a Unique Games gap instance for the basic semidefinite program strengthened by all valid linear inequalities on the inner products of up to exp(Ω(log log n) 1/4) vectors. For a stronger relaxation obtained from the basic semidefinite program by R rounds of Sherali–Adams liftandproject, we prove a Unique Games integrality gap for R = Ω(log log n) 1/4. By composing these SDP gaps with UGChardness reductions, the above results imply corresponding integrality gaps for every problem for which a UGCbased hardness is known. Consequently, this work implies that including any valid constraints on up to exp(Ω(log log n) 1/4) vectors to natural semidefinite program, does not improve the approximation ratio for any problem in the following classes: constraint satisfaction problems, ordering constraint satisfaction problems and metric labeling problems over constantsize metrics. We obtain similar SDP integrality gaps for Balanced Separator, building on [11]. We also exhibit, for explicit constants γ, δ> 0, an npoint negativetype metric which requires distortion Ω(log log n) γ to embed into ℓ1, although all its subsets of size exp(Ω(log log n) δ) embed isometrically into ℓ1. Keywordssemidefinite programming, approximation algorithms, unique games conjecture, hardness of approximation, SDP hierarchies, Sherali–Adams hierarchy, integrality gap construction 1.
Improved lower bounds for embeddings into L1
 SIAM J. COMPUT.
, 2009
"... We improve upon recent lower bounds on the minimum distortion of embedding certain finite metric spaces into L1. In particular, we show that for every n ≥ 1, there is an npoint metric space of negative type that requires a distortion of Ω(log log n) for such an embedding, implying the same lower bo ..."
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Cited by 40 (5 self)
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We improve upon recent lower bounds on the minimum distortion of embedding certain finite metric spaces into L1. In particular, we show that for every n ≥ 1, there is an npoint metric space of negative type that requires a distortion of Ω(log log n) for such an embedding, implying the same lower bound on the integrality gap of a wellknown semidefinite programming relaxation for sparsest cut. This result builds upon and improves the recent lower bound of (log log n) 1/6−o(1) due to Khot and Vishnoi [The unique games conjecture, integrality gap for cut problems and the embeddability of negative type metrics into l1, in Proceedings of the 46th Annual IEEE Symposium
A brief introduction to Fourier analysis on the Boolean cube
 Theory of Computing Library– Graduate Surveys
, 2008
"... Abstract: We give a brief introduction to the basic notions of Fourier analysis on the ..."
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Cited by 34 (4 self)
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Abstract: We give a brief introduction to the basic notions of Fourier analysis on the
On the Fourier tails of bounded functions over the discrete cube
 IN PROC
, 2006
"... In this paper we consider bounded realvalued functions over the discrete cube, f: {−1, 1} n → [−1, 1]. Such functions arise naturally in theoretical computer science, combinatorics, and the theory of social choice. It is often interesting to understand when these functions essentially depend on few ..."
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Cited by 34 (4 self)
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In this paper we consider bounded realvalued functions over the discrete cube, f: {−1, 1} n → [−1, 1]. Such functions arise naturally in theoretical computer science, combinatorics, and the theory of social choice. It is often interesting to understand when these functions essentially depend on few coordinates. Our main result is a dichotomy that includes a lower bound on how fast the Fourier coefficients of such functions can decay: we show that S>k �f(S) 2 ≥ exp(−O(k 2 log k)), unless f depends essentially on only 2 O(k) coordinates. We also show, perhaps surprisingly, that this result is sharp up to the log k factor. The same type of result has already been proven (and shown useful) for Boolean functions [Bou02, KS]. The proof of these results relies on the Booleanity of the functions, and does not generalize to all bounded functions. In this work we handle all bounded functions, at the price
Graph Products, Fourier Analysis and Spectral Techniques
, 2003
"... We consider powers of regular graphs defined by the weak graph product and give a characterization of maximumsize independent sets for a wide family of base graphs which includes, among others, complete graphs, line graphs of regular graphs which contain a perfect matching and Kneser graphs. In man ..."
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Cited by 33 (10 self)
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We consider powers of regular graphs defined by the weak graph product and give a characterization of maximumsize independent sets for a wide family of base graphs which includes, among others, complete graphs, line graphs of regular graphs which contain a perfect matching and Kneser graphs. In many cases this also characterizes the optimal colorings of these products. We show that the independent sets induced by the base graph are the only maximumsize independent sets. Furthermore we give a qualitative stability statement: any independent set of size close to the maximum is close to some independent set of maximum size. Our approach is based on Fourier analysis on Abelian groups and on Spectral Techniques. To this end we develop some basic lemmas regarding the Fourier transform of functions on f0; : : : ; r \Gamma 1gn, generalizing some useful results from the f0; 1gn case.
On nonapproximability for quadratic programs
 IN 46TH ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 2005
"... This paper studies the computational complexity of the following type of quadratic programs: given an arbitrary matrix whose diagonal elements are zero, find x ∈ {−1, 1} n that maximizes x T Mx. This problem recently attracted attention due to its application in various clustering settings, as well ..."
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Cited by 30 (4 self)
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This paper studies the computational complexity of the following type of quadratic programs: given an arbitrary matrix whose diagonal elements are zero, find x ∈ {−1, 1} n that maximizes x T Mx. This problem recently attracted attention due to its application in various clustering settings, as well as an intriguing connection to the famous Grothendieck inequality. It is approximable to within a factor of O(log n), and known to be NPhard to approximate within any factor better than 13/11 − ɛ for all ɛ> 0. We show that it is quasiNPhard to approximate to a factor better than O(log γ n) for some γ> 0. The integrality gap of the natural semidefinite relaxation for this problem is known as the Grothendieck constant of the complete graph, and known to be Θ(log n). The proof of this fact was nonconstructive, and did not yield an explicit problem instance where this integrality gap is achieved. Our techniques yield an explicit instance for which the integrality gap is log n Ω ( log log n), essentially answering one of the open problems of Alon et al. [AMMN].