## Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? (2005)

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Citations: | 173 - 26 self |

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@MISC{Khot05optimalinapproximability,

author = {Subhash Khot and Guy Kindler and Elchanan Mossel and Ryan O'Donnell},

title = {Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? },

year = {2005}

}

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### Abstract

In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the Goemans-Williamson algorithm [25]. This implies that if the Unique Games

### Citations

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Citation Context ... regime would be a function of the form f : [q] n → [q]. However, as we did for boolean functions, we will consider a continuous relaxation of the range. Specifically, define ∆q = {(x1, . . . , xq) ∈ =-=[0, 1]-=- q : � xi = 1}, which can be thought of as the space of probability distributions over [q]. We will consider functions f : [q] n → ∆q; this generalizes functions f : [q] n → [q] if we identify the ele... |

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Citation Context ...following versions of the Central Limit Theorem with error bounds: the first is a multidimensional version from [5, Corollary 16.3]; the second is the (non-uniform) version of the BerryEsseen theorem =-=[20]-=-: Theorem 16. Let X1, . . . , Xn be independent random variables taking values in R k satisfying: • E[Xj] = 0, j = 1 . . . n; • n −1 � n j=1 Cov(Xj) = V , where Cov denotes the variance-covariance mat... |

1429 |
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Citation Context ...of the problem in which each edge is assigned a nonnegative weight and the goal is to cut as much weight as possible. MAX-CUT is NP-complete (indeed, it is one of Karp’s original NP-complete problems =-=[23]-=-) and so it is of interest to try to find polynomial time approximation algorithms. For maximization problems such as MAX-CUT we say an algorithm gives an α-approximation if it always returns an answe... |

938 | Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming
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Citation Context ...er we give evidence suggesting that MAXCUT is NP-hard to approximate to within a factor of αGW+ɛ, for all ɛ > 0, where αGW denotes the approximation ratio achieved by the Goemans-Williamson algorithm =-=[14]-=-, αGW ≈ .878567. This result is conditional, relying on two conjectures: a) the Unique Games conjecture of Khot [24]; and, b) a very believable conjecture we call the Majority Is Stablest conjecture. ... |

647 | Some optimal inapproximability results
- Håstad
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Citation Context ... to this last point. Since the Goemans-Williamson algorithm was published a decade ago there has been no algorithmic progress on approximating MAX-CUT. Since H˚astad’s classic inapproximability paper =-=[17]-=- from two years later there has been no progress on the hardness of approximating MAX-CUT, except for the creation of a better reduction gadget. As one of the most natural and simple problems to have ... |

571 |
Optimization, approximation and complexity classes
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Citation Context ...lgorithm might be considered surprising, but as we shall see, this geometry seems to be an inherent part of the MAX-CUT problem. On the hardness of approximation side, MAX-CUT was proved MAX-SNP hard =-=[26]-=- and Bellare, Goldreich, and Sudan [1] explicitly showed that it was NP-hard to approximate MAX-CUT to any factor higher than 83/84. The hardness factor was improved to 16/17 ≈ .941176 by H˚astad [19]... |

277 |
P-complete approximation problems
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Citation Context ...ed algorithm for MAX-CUT — put each vertex on either side of the partition independently with equal probability — is a 1/2-approximation, and this algorithm is easy to derandomize; Sahni and Gonzalez =-=[27]-=- gave the first 1/2-approximation algorithm in 1976. Following this some (1/2 + o(1))-approximation algorithms were given, but no real progress was made until the breakthrough 1994 paper of Goemans an... |

232 | On the power of unique 2-prover 1-round games
- Khot
- 2002
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Citation Context ...re αGW denotes the approximation ratio achieved by the Goemans-Williamson algorithm [14], αGW ≈ .878567. This result is conditional, relying on two conjectures: a) the Unique Games conjecture of Khot =-=[24]-=-; and, b) a very believable conjecture we call the Majority Is Stablest conjecture. These results indicate that the geometric nature of the Goemans-Williamson algorithm might be intrinsic to the MAX-C... |

224 | The influence of variables on boolean functions, FOCS
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Citation Context ...PCP-based inapproximability results. Second, the conjecture is an extension of, or is very similar to, several other important theorems in the analysis of boolean functions, including the KKL theorem =-=[20]-=- and Bourgain’s theorem [3]; and furthermore, the partial progress we make on proving the Majority Is Stablest conjecture clarifies certain aspects of the papers of Friedgut, Kalai, and Naor [13] (cf.... |

205 | Free bits, PCPs, and nonapproximability— towards tight results
- Bellare, Goldreich, et al.
- 1998
(Show Context)
Citation Context ..., but as we shall see, this geometry seems to be an inherent part of the MAX-CUT problem. On the hardness of approximation side, MAX-CUT was proved MAX-SNP hard [26] and Bellare, Goldreich, and Sudan =-=[1]-=- explicitly showed that it was NP-hard to approximate MAX-CUT to any factor higher than 83/84. The hardness factor was improved to 16/17 ≈ .941176 by H˚astad [19] via a reduction from MAX-3LIN using a... |

161 | Improved Approximation Algorithms for Max k-cut and Max Bisection.", Integer Programming and Combinatorial Optimization - Frieze, Jerrum - 1995 |

132 | Approximating the value of two prover proof systems, with applications to MAX
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Citation Context ... constraints to the semidefinite program has been suggested, alternate rounding schemes have been proposed, and local modification heuristics that work for special graphs have been proven (see, e.g., =-=[14, 9, 8, 22, 32, 10, 11]-=-). And of course, perhaps a completely different algorithm altogether can perform better. Several papers have either explicitly ([8]) or implicitly ([11]) given the problem of improving on αGW as an i... |

131 | Every monotone graph property has a sharp threshold
- Friedgut, Kalai
- 1996
(Show Context)
Citation Context ...en a long line of work in the analysis of boolean functions studying the noise sensitivity of functions and the associated Fourier-theoretic quantities (some examples, roughly in chronological order: =-=[33, 8, 52, 23, 53, 9, 22, 4, 6, 7, 24, 34, 44, 46, 13]-=-). Building on the intuition gathered from this past work, we were motivated to make the Majority Is Stablest conjecture in the originial version of the paper. We discuss these relevant previous resul... |

126 | The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative Type Metrics into l 1 .InFOCS 2005
- Khot, Vishnoi
- 2005
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Citation Context ...ariable/equation structure need not be bipartite. But in fact, it is easy to observe that the “non-bipartite” version of the Unique Games Conjecture is equivalent to the usual Unique Games Conjecture =-=[39]-=- (up to a factor of 2 in the soundness). Hence Theorem 12 and its corollaries may be viewed as concerning the allowable parameter tradeoffs in the Unique Games Conjecture. In particular, Corollary 13 ... |

121 | Normal Approximation and Asymptotic Expansions - Bhattacharya, Rao - 1976 |

117 | Vertex Cover Might be Hard to Approximate to within 2
- Khot, Regev
- 2003
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Citation Context ...ion into the Unique Games conjecture via the lens of MAX-CUT. First, we see that the conjecture does not give an incorrectly strong hardness bound for MAX-CUT, and indeed (as it does for Vertex Cover =-=[25]-=-) it gives what would ultimately be a natural bound. Second, we show (by reduction) that (modulo the Majority Is Stablest conjecture) the Unique Games problem is not harder than the problem of beating... |

88 |
Collective Coin-Flipping
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Citation Context ...what follows we consider the set of strings {−1, 1} n to be a probability space under the uniform distribution. First we recall the well-known notion of ‘influence’, introduced to computer science in =-=[3]-=- and studied even earlier in economics. Definition 2. Let f : {−1, 1} n → R. Then the influence of xi on f is defined by Infi(f) = E [Varxi [f]] . (x1,...,xi−1,xi+1,...,xn) 5s(Note that for f : {−1, 1... |

88 | Noise stability of functions with low influences: invariance and optimality
- Mossel, O’Donnell, et al.
- 2005
(Show Context)
Citation Context ...ic to the MAX-CUT problem. Our reduction relies on a theorem we call Majority Is Stablest. This was introduced as a conjecture in the original version of this paper, and was subsequently confirmed in =-=[45]-=-. A stronger version of this conjecture called Plurality Is Stablest is still open, although [45] contains a proof of an asymptotic version of it. Our techniques extend to several other two-variable c... |

85 |
A PCP characterization of NP with optimal amortized query complexity
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(Show Context)
Citation Context ...ables via a binary code and then running PCP tests on the (supposed) encodings. This approach has been immensely successful in proving inapproximability results for k-CSPs with k ≥ 3 (see for example =-=[19, 28, 16]-=-). However the approach gets stuck in the case of 2-CSPs. We seem to have no techniques for constructing boolean 2-query PCPs and the bottleneck seems to be the lack of an appropriate PCP ‘outer verif... |

84 | The importance of being biased
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- 2002
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Citation Context ...ture implies that Vertex Cover is NP-hard to approximate within any factor less than 2. The inner verifier in their paper is based on Friedgut’s theorem and is inspired by the work of Dinur and Safra =-=[7]-=- that showed 1.36 hardness for Vertex Cover. In the present paper we continue this line of research and propose a plausible direction for attacking the MAX-CUT problem. We do construct an inner verifi... |

71 | Noise sensitivity of Boolean functions and applications to percolation, Publ
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Citation Context ... bit, and seeing if this changes the value of f, we can instead flip a constant fraction (in expectation) of the bits. This leads to the study of ‘noise sensitivity’, pioneered in computer science by =-=[20, 17, 4]-=-. Definition 3. Let f : {−1, 1} n → R and let −1 ≤ ρ ≤ 1. The noise correlation of f at ρ is defined as follows: Let x be a uniformly random string in {−1, 1} n and let y be a ‘ρcorrelated’ copy; i.e.... |

62 |
Boolean functions with low average sensitivity depend on few coordinates
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Citation Context ...desired inapproximability results. However the inner verifier typically relies on rather deep theorems about the Fourier spectrum of boolean functions, e.g. the theorem of Bourgain [3] or of Friedgut =-=[12]-=-. The Unique Games conjecture was used in [24] to show that Min-2SAT-Deletion is NP-hard to approximate within any constant factor. The inner verifier is based on a test proposed by H˚astad [18] and o... |

60 | Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to Max-Cut and other problems
- Zwick
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Citation Context ... constraints to the semidefinite program has been suggested, alternate rounding schemes have been proposed, and local modification heuristics that work for special graphs have been proven (see, e.g., =-=[14, 9, 8, 22, 32, 10, 11]-=-). And of course, perhaps a completely different algorithm altogether can perform better. Several papers have either explicitly ([8]) or implicitly ([11]) given the problem of improving on αGW as an i... |

49 |
The influence of variables in product spaces
- Bourgain, Kahn, et al.
- 1992
(Show Context)
Citation Context ...en a long line of work in the analysis of boolean functions studying the noise sensitivity of functions and the associated Fourier-theoretic quantities (some examples, roughly in chronological order: =-=[33, 8, 52, 23, 53, 9, 22, 4, 6, 7, 24, 34, 44, 46, 13]-=-). Building on the intuition gathered from this past work, we were motivated to make the Majority Is Stablest conjecture in the originial version of the paper. We discuss these relevant previous resul... |

43 |
On the distribution of the fourier spectrum of boolean functions
- Bourgain
(Show Context)
Citation Context ...results. Second, the conjecture is an extension of, or is very similar to, several other important theorems in the analysis of boolean functions, including the KKL theorem [20] and Bourgain’s theorem =-=[3]-=-; and furthermore, the partial progress we make on proving the Majority Is Stablest conjecture clarifies certain aspects of the papers of Friedgut, Kalai, and Naor [13] (cf. Theorem 8) and Talagrand [... |

43 | On the optimality of the random hyperplane rounding technique for MAX CUT. Random Structures and Algorithms
- Feige, Schechtman
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Citation Context ...programmingbased algorithms, no one has been able to improve on the algorithm of Goemans and Williamson. Although the true approximation ratio of Goemans-Williamson was proved to be not more than αGW =-=[22, 11]-=- and the integrality gap of their semidefinite relaxation was also proved to be αGW [11], there appears on the face of it to be plenty of possibilities for improvement. Adding triangle constraints and... |

37 | Conditional hardness for approximate coloring
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Citation Context ...lest theorem has interesting applications outside of this work — to the economic theory of social choice [34] for example — and has already proven useful for other PCP-based inapproximability results =-=[14]-=-. In Section 6.3 we mention interesting generalizations of the Majority Is Stablest theorem for q-ary functions, q > 2, which are relevant for hardness of approximation and are not resolved in full. D... |

36 |
Improved rounding techniques for the MAX 2-SAT and MAX DI-CUT problems
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Citation Context ... for MAX-2SAT, MAX-q-CUT, and MAX-2LIN(q). For MAX-2SAT we show approximation hardness up to a factor of roughly .943. This nearly matches the .940 approximation algorithm of Lewin, Livnat, and Zwick =-=[41]-=-. Furthermore, we show that our .943... factor is actually tight for a slightly restricted version of MAX-2SAT. For MAX-q-CUT we show a hardness factor which asymptotically (for large q) matches the a... |

33 |
How good is the GoemansWilliamson MAX CUT algorithm
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Citation Context ...programmingbased algorithms, no one has been able to improve on the algorithm of Goemans and Williamson. Although the true approximation ratio of Goemans-Williamson was proved to be not more than αGW =-=[22, 11]-=- and the integrality gap of their semidefinite relaxation was also proved to be αGW [11], there appears on the face of it to be plenty of possibilities for improvement. Adding triangle constraints and... |

33 | Approximation algorithms for unique games
- Trevisan
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Citation Context ...ks on the Unique Games Conjecture, which if passed will refute it. 2sIndeed, works subsequent to the original version of this paper have provided approximation algorithms for the Unique Games problem =-=[54, 29, 10]-=- improving on Khot’s original algorithm [37]. In particular, in [10] Charikar, Makarychev, and Makarychev gave a semidefinite programming-based approximation algorithm for Unique Games whose approxima... |

31 | Hardness of approximate hypergraph coloring
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Citation Context ...ables via a binary code and then running PCP tests on the (supposed) encodings. This approach has been immensely successful in proving inapproximability results for k-CSPs with k ≥ 3 (see for example =-=[19, 28, 16]-=-). However the approach gets stuck in the case of 2-CSPs. We seem to have no techniques for constructing boolean 2-query PCPs and the bottleneck seems to be the lack of an appropriate PCP ‘outer verif... |

29 |
linear programming
- Gadgets
(Show Context)
Citation Context ...te MAX-CUT to any factor higher than 83/84. The hardness factor was improved to 16/17 ≈ .941176 by H˚astad [19] via a reduction from MAX-3LIN using a gadget of Trevisan, Sorkin, Sudan, and Williamson =-=[31]-=-. This stands as the current best hardness result. Despite much effort and many improvements in the approximation guarantees of other semidefinite programmingbased algorithms, no one has been able to ... |

28 | Approximation algorithms for max-3-cut and other problems via complex semidefinite programming - Goemans, Williamson - 2004 |

27 | Influences of variables and threshold intervals under group symmetries, Geom
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Citation Context ...em 8 for unbalanced functions: known techniques in bounding Fourier weights at low levels first apply random restrictions, and then bound the Fourier weight at level 1 of the resulting function; c.f. =-=[5, 30, 3]-=-. When performing a random restriction of a balanced function, the resulting function may be unbalanced. We give the following generalization of Theorem 8, which essentially also generalizes the theor... |

24 | Boolean Functions whose Fourier Transform is Concentrated on the First Two Levels Adv
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Citation Context ...orem [20] and Bourgain’s theorem [3]; and furthermore, the partial progress we make on proving the Majority Is Stablest conjecture clarifies certain aspects of the papers of Friedgut, Kalai, and Naor =-=[13]-=- (cf. Theorem 8) and Talagrand [30] (cf. Theorem 10). We note that our partial progress lets us prove an inapproximability factor of .909155 for MAX-CUT assuming only the Unique Games conjecture; this... |

22 | Approximating unique games
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Citation Context ...ks on the Unique Games Conjecture, which if passed will refute it. 2sIndeed, works subsequent to the original version of this paper have provided approximation algorithms for the Unique Games problem =-=[54, 29, 10]-=- improving on Khot’s original algorithm [37]. In particular, in [10] Charikar, Makarychev, and Makarychev gave a semidefinite programming-based approximation algorithm for Unique Games whose approxima... |

20 |
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Citation Context ...]; and furthermore, the partial progress we make on proving the Majority Is Stablest conjecture clarifies certain aspects of the papers of Friedgut, Kalai, and Naor [13] (cf. Theorem 8) and Talagrand =-=[30]-=- (cf. Theorem 10). We note that our partial progress lets us prove an inapproximability factor of .909155 for MAX-CUT assuming only the Unique Games conjecture; this is already stronger than the best ... |

20 | On the Fourier tails of bounded functions over the discrete cube - Dinur, Friedgut, et al. |

19 | On Weighted vs Unweighted Versions of Combinatorial Optimization Problems
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Citation Context ...n answer which is at least α times the optimal value; we also often relax this definition to allow randomized algorithms which in expectation give α-approximations. Crescenzi, Silvestri, and Trevisan =-=[6]-=- have shown that the weighted and unweighted versions of MAX-CUT have equal optimal approximation factors (up to an additive o(1)) and so we pass freely between the two problems in this paper.sThe tri... |

19 | Improved approximation of max-cut on graphs of bounded degree
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(Show Context)
Citation Context ... constraints to the semidefinite program has been suggested, alternate rounding schemes have been proposed, and local modification heuristics that work for special graphs have been proven (see, e.g., =-=[14, 9, 8, 22, 32, 10, 11]-=-). And of course, perhaps a completely different algorithm altogether can perform better. Several papers have either explicitly ([8]) or implicitly ([11]) given the problem of improving on αGW as an i... |

19 | Every linear threshold function has a low-weight approximator - Servedio |

17 |
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Citation Context ...en a long line of work in the analysis of boolean functions studying the noise sensitivity of functions and the associated Fourier-theoretic quantities (some examples, roughly in chronological order: =-=[33, 8, 52, 23, 53, 9, 22, 4, 6, 7, 24, 34, 44, 46, 13]-=-). Building on the intuition gathered from this past work, we were motivated to make the Majority Is Stablest conjecture in the originial version of the paper. We discuss these relevant previous resul... |

16 | WARNERS: On approximate graph colouring and MAX-k-CUT algorithms based on the ϑ-function
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Citation Context ...Sρ(f) ≤ PlurStab(q, ρ) + ɛ Note that in the case q = 2, Sheppard’s formula gives PlurStab(2, ρ) = 1 − 2 π arccos ρ, which is the noise stability of Majority; there is also a closed formula for q = 3 (=-=[27, 12]-=-). For large values of q we give asymptotics which hold up to a 1 + oq(1) factor in Section 6. For the reader’s convenience, we remark here that PlurStab(q, ρ) = ˜ � Θ (1/q) (1−ρ)/(1+ρ)� . Although we... |

15 |
On systems of linear equations with two variables per equation
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- 2004
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Citation Context ...aints is 1−η, their semidefinite programming algorithm produces a solution satisfying about a fraction (1/q) η/(2−3η) . The best known NPhardness results come from a recent work of Feige and Reichman =-=[21]-=-. They show that it is NP-hard to approximate MAX-2LIN(q) to within a factor of 1/q β for some universal constant β > 0; however this hardness is located at a gap of ɛ vs. ɛ/q β . In particular, given... |

13 |
On the application of the theory of errors to cases of normal distribution and normal correlation
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Citation Context ...approaches (1 − 2 π arccos ρ) (this fact was stated in a paper of Gulibaud from the 1960’s [15] and is ultimately derived from the Central Limit theorem plus a result from an 1890’s paper of Sheppard =-=[29]-=-). Thus we have the formal statement of the conjecture: Majority Is Stablest conjecture: Fix ρ ∈ [0, 1). Then for any ɛ > 0 there is a small enough δ = δ(ɛ, ρ) > 0 such that if f : {−1, 1} n → [−1, 1]... |

10 |
An appendix to Sharp thresholds of graph properties, and the k-SAT problem, by
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9 |
A Fourier-theoretic perspective on the Concordet paradox and Arrow’s theorem
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Citation Context ...erts in “analysis of boolean functions” whom we have consulted have agreed that the conjecture should be correct (in fact, similar conjectures already appear in the literature, e.g. Conjecture 5.1 in =-=[21]-=-), and every relevant piece of evidence is in concordance with the conjecture. Because of this, we believe that understanding the status of the Unique Games conjecture is the main issue. Unlike the Ma... |

8 |
Improved approximation algorithms for
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Citation Context ...actually tight for a slightly restricted version of MAX-2SAT. For MAX-q-CUT we show a hardness factor which asymptotically (for large q) matches the approximation factor achieved by Frieze and Jerrum =-=[25]-=-, namely 1 − 1/q + 2(ln q)/q 2 . For MAX-2LIN(q) we show hardness of distinguishing between instances which are (1−ɛ)-satisfiable and those which are not even, roughly, (q −ɛ/2 )-satisfiable. These pa... |

8 | On the noise sensitivity of monotone functions
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Citation Context |

7 |
Theories of the general interest, and the logical problem of aggregation, in
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Citation Context ..., for every ρ. Note that when n tends to infinity, the noise correlation at ρ of the n-bit Majority function approaches (1 − 2 π arccos ρ) (this fact was stated in a paper of Gulibaud from the 1960’s =-=[15]-=- and is ultimately derived from the Central Limit theorem plus a result from an 1890’s paper of Sheppard [29]). Thus we have the formal statement of the conjecture: Majority Is Stablest conjecture: Fi... |