Results 1  10
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12
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 175 (26 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
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 q = ..."
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Cited by 37 (12 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].
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 22 (3 self)
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Abstract: We give a brief introduction to the basic notions of Fourier analysis on the
On the Noise Sensitivity of Monotone Functions
, 2003
"... It is known that for all monotone functions f: {0, 1} n → {0, 1}, if x ∈ {0, 1} n is chosen uniformly at random and y is obtained from x by flipping each of the bits of x independently with probability ɛ = n −α, then P[f(x) � = f(y)] < cn −α+1/2, for some c> 0. Previously, the best construction of m ..."
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Cited by 8 (4 self)
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It is known that for all monotone functions f: {0, 1} n → {0, 1}, if x ∈ {0, 1} n is chosen uniformly at random and y is obtained from x by flipping each of the bits of x independently with probability ɛ = n −α, then P[f(x) � = f(y)] < cn −α+1/2, for some c> 0. Previously, the best construction of monotone functions satisfying P[fn(x) � = fn(y)] ≥ δ, where 0 < δ < 1/2, required ɛ ≥ c(δ)n −α, where α = 1 − ln 2 / ln 3 = 0.36907..., and c(δ)> 0. We improve this result by achieving for every 0 < δ < 1/2, P[fn(x) � = fn(y)] ≥ δ, with: • ɛ = c(δ)n−α for any α < 1/2, using the recursive majority function with arity k = k(α); π/2 =.3257..., using an explicit recursive majority • ɛ = c(δ)n −1/2 log t n for t = log 2 function with increasing arities; and, • ɛ = c(δ)n −1/2, nonconstructively, following a probabilistic CNF construction due to Talagrand. We also study the problem of achieving the best dependence on δ in the case that the noise rate ɛ is at least a small constant; the results we obtain are tight to within logarithmic factors.
Edgeisoperimetric inequalities and influences
 In Combinatorics, Probability, and Computing
, 2006
"... Abstract We give a combinatorial proof of the result of Kahn, Kalai, and Linial [19], which statesthat every balanced boolean function on the ndimensional boolean cube has a variable with influence of at least \Omega i log nn j. The methods of the proof are then used to recover additional isoperime ..."
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Cited by 8 (1 self)
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Abstract We give a combinatorial proof of the result of Kahn, Kalai, and Linial [19], which statesthat every balanced boolean function on the ndimensional boolean cube has a variable with influence of at least \Omega i log nn j. The methods of the proof are then used to recover additional isoperimetric results forthe cube, with improved constants. We also state some conjectures about optimal constants and discuss their possible implications.
Noninteractive correlation distillation, inhomogeneous Markov chains, and the reverse BonamiBeckner inequality
 Israel Journal of Mathematics
"... In this paper we study noninteractive correlation distillation (NICD), a generalization of noise sensitivity previously considered in [5, 31, 39]. We extend the model to NICD on trees. In this model there is a fixed undirected tree with players at some of the nodes. One node is given a uniformly ra ..."
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Cited by 5 (2 self)
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In this paper we study noninteractive correlation distillation (NICD), a generalization of noise sensitivity previously considered in [5, 31, 39]. We extend the model to NICD on trees. In this model there is a fixed undirected tree with players at some of the nodes. One node is given a uniformly random string and this string is distributed throughout the network, with the edges of the tree acting as independent binary symmetric channels. The goal of the players is to agree on a shared random bit without communicating. Our new contributions include the following: • In the case of a kleaf star graph (the model considered in [31]), we resolve the open question of whether the success probability must go to zero as k → ∞. We show that this is indeed the case and provide matching upper and lower bounds on the asymptotically optimal rate (a slowlydecaying polynomial). • In the case of the kvertex path graph, we show that it is always optimal for all players to use the same 1bit function. • In the general case we show that all players should use monotone functions. We also show, somewhat
Influences in Product Spaces: KKL and BKKKL Revisited
"... The notion of the influence of a variable on a Boolean function on a product space has drawn much attention in combinatorics, computer science and other fields. Two of the basic papers dealing with this notion are [KKL] and [BKKKL]. ..."
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Cited by 1 (0 self)
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The notion of the influence of a variable on a Boolean function on a product space has drawn much attention in combinatorics, computer science and other fields. Two of the basic papers dealing with this notion are [KKL] and [BKKKL].
unknown title
"... years and there is now a significant body of research on the subject (for a survey, see e.g., [8]). The second topic is variable influences, introduced to theoretical computer science by BenOr and Linial in 1985 [2]. Any nvariate boolean function f has an associated influence vector (Inf1(f),...,I ..."
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years and there is now a significant body of research on the subject (for a survey, see e.g., [8]). The second topic is variable influences, introduced to theoretical computer science by BenOr and Linial in 1985 [2]. Any nvariate boolean function f has an associated influence vector (Inf1(f),...,Inf n(f)) where Infi(f) measures the extent to which the value of f depends on variable i (a precise definition appears in Section 1.2). A number of papers have dealt with properties of this vector and its relation to other properties of boolean functions; perhaps the best known work along these lines is that of Kahn, Kalai and Linial [14] (“KKL”) concerning the maximum influenceInfmax(f) = max{Infi(f): i ∈ [n]}. Their result implies, for example, that log n
Submitted to the Senate of TelAviv University
"... under the supervision of prof. Shmuel Safra ..."