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10
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 173 (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
Gaussian Bounds for Noise Correlation of Functions and Tight Analysis of Long Codes
 In IEEE Symposium on Foundations of Computer Science (FOCS
, 2008
"... In this paper we derive tight bounds on the expected value of products of low influence functions defined on correlated probability spaces. The proofs are based on extending Fourier theory to an arbitrary number of correlated probability spaces, on a generalization of an invariance principle recentl ..."
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Cited by 35 (5 self)
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In this paper we derive tight bounds on the expected value of products of low influence functions defined on correlated probability spaces. The proofs are based on extending Fourier theory to an arbitrary number of correlated probability spaces, on a generalization of an invariance principle recently obtained with O’Donnell and Oleszkiewicz for multilinear polynomials with low influences and bounded degree and on properties of multidimensional Gaussian distributions. We present two applications of the new bounds to the theory of social choice. We show that Majority is asymptotically the most predictable function among all low influence functions given a random sample of the voters. Moreover, we derive an almost tight bound in the context of Condorcet aggregation and low influence voting schemes on a large number of candidates. In particular, we show that for every low influence aggregation function, the probability that Condorcet voting on k candidates will result in a unique candidate that is preferable to all others is k−1+o(1). This matches the asymptotic behavior of the majority function for which the probability is k−1−o(1). A number of applications in hardness of approximation in theoretical computer science were
Maximally stable Gaussian partitions with discrete applications
 Israel J. Math
"... Gaussian noise stability results have recently played an important role in proving results in hardness of approximation in computer science and in the study of voting schemes in social choice. We prove a new Gaussian noise stability result generalizing an isoperimetric result by Borell on the heat k ..."
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Cited by 4 (0 self)
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Gaussian noise stability results have recently played an important role in proving results in hardness of approximation in computer science and in the study of voting schemes in social choice. We prove a new Gaussian noise stability result generalizing an isoperimetric result by Borell on the heat kernel and derive as applications: • An optimality result for majority in the context of Condorcet voting. • A proof of a conjecture on “cosmic coin tossing ” for low influence functions. We also discuss a Gaussian noise stability conjecture which may be viewed as a generalization of the “Double Bubble ” theorem and show that it implies: • A proof of the “Plurality is Stablest Conjecture”. • That the FriezeJerrum SDP for MAXqCUT achieves the optimal approximation factor assuming the Unique Games Conjecture.
New Maximally Stable Gaussian Partitions with Discrete Applications
, 2009
"... Gaussian noise stability results have recently played an important role in proving fundamental results in hardness of approximation in computer science and in the study of voting schemes in social choice. We propose two Gaussian noise stability conjectures and derive consequences of the conjectures ..."
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Cited by 2 (0 self)
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Gaussian noise stability results have recently played an important role in proving fundamental results in hardness of approximation in computer science and in the study of voting schemes in social choice. We propose two Gaussian noise stability conjectures and derive consequences of the conjectures in hardness of approximation and social choice. Both conjectures generalize isoperimetric results by Borell on the heat kernel. One of the conjectures may be also be viewed as a generalization of the ”Double Bubble ” theorem. The applications of the conjectures include an optimality result for majority in the context of Condorcet voting and a proof that the FriezeJerrum SDP for MAXqCUT achieves the optimal approximation factor assuming the Unique Games Conjecture. We finally derive a short proof of the first conjecture based on the extended Riesz inequality. 1
A Quantitative Arrow Theorem
, 903
"... Arrow’s Impossibility Theorem states that any constitution which satisfies Independence of Irrelevant Alternatives (IIA) and Unanimity and is not a Dictator has to be nontransitive. In this paper we study quantitative versions of Arrow theorem. Consider n voters who vote independently at random, ea ..."
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Cited by 1 (0 self)
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Arrow’s Impossibility Theorem states that any constitution which satisfies Independence of Irrelevant Alternatives (IIA) and Unanimity and is not a Dictator has to be nontransitive. In this paper we study quantitative versions of Arrow theorem. Consider n voters who vote independently at random, each following the uniform distribution over the 6 rankings of 3 alternatives. Arrow’s theorem implies that any constitution which satisfies IIA and Unanimity and is not a dictator has a probability of at least 6 −n for a nontransitive outcome. When n is large, 6 −n is a very small probability, and the question arises if for large number of voters it is possible to avoid paradoxes with probability close to 1. Here we give a negative answer to this question by proving that for every ǫ> 0, there exists a δ = δ(ǫ)> 0, which depends on ǫ only, such that for all n, and all constitutions on 3 alternatives, if the constitution satisfies: • The IIA condition. • For every pair of alternatives a, b, the probability that the constitution ranks a above b is at least ǫ.
Arrow’s Impossibility Without Unanimity
, 2009
"... Arrow’s Impossibility Theorem states that any constitution which satisfies Transitivity, Independence of Irrelevant Alternatives (IIA) and Unanimity is a dictatorship. Wilson derived properties of constitutions satisfying Transitivity and IIA for unrestricted domains where ties are allowed. In this ..."
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Arrow’s Impossibility Theorem states that any constitution which satisfies Transitivity, Independence of Irrelevant Alternatives (IIA) and Unanimity is a dictatorship. Wilson derived properties of constitutions satisfying Transitivity and IIA for unrestricted domains where ties are allowed. In this paper we consider the case where only strict preferences are allowed. In this case we derive a new short proof of Arrow theorem and further obtain a new and complete characterization of all functions satisfying Transitivity and IIA. The proof is based on a variant of the method of pivotal voters due to Barbera. 1
unknown title
"... Noise stability of functions with low influences: invariance and optimality In this paper we study functions with low influences on product probability spaces. The analysis of boolean functions f: {−1, 1} n → {−1, 1} with low influences has become a central problem in discrete Fourier analysis. It i ..."
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Noise stability of functions with low influences: invariance and optimality In this paper we study functions with low influences on product probability spaces. The analysis of boolean functions f: {−1, 1} n → {−1, 1} with low influences has become a central problem in discrete Fourier analysis. It is motivated by fundamental questions arising from the construction of probabilistically checkable proofs in theoretical computer science and from problems in the theory of social choice in economics. We prove an invariance principle for multilinear polynomials with low influences and bounded degree; it shows that under mild conditions the distribution of such polynomials is essentially invariant for all product spaces. Ours is one of the very few known nonlinear invariance principles. It has the advantage that its proof is simple and that the error bounds are explicit. We also show that the assumption of bounded degree can be eliminated if the polynomials are slightly “smoothed”; this extension is essential for our applications to “noise stability”type problems. In particular, as applications of the invariance principle we prove two conjectures: the “Majority Is Stablest ” conjecture [27] from theoretical computer science, which was the original motivation for this work, and the “It Ain’t Over Till It’s Over ” conjecture [25] from social choice theory. The “Majority Is Stablest ” conjecture and its generalizations proven here in conjunction with “Unique Games ” and its variants imply a number of (optimal) inapproximability results for graph problems. 1
Complete Characterization of Functions Satisfying the Conditions of Arrow’s Theorem
, 910
"... Arrow’s theorem implies that a social welfare function satisfying Transitivity, the Weak Pareto Principle (Unanimity) and Independence of Irrelevant Alternatives (IIA) must be dictatorial. When nonstrict preferences are also allowed, a dictatorial social welfare function is defined as a function fo ..."
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Arrow’s theorem implies that a social welfare function satisfying Transitivity, the Weak Pareto Principle (Unanimity) and Independence of Irrelevant Alternatives (IIA) must be dictatorial. When nonstrict preferences are also allowed, a dictatorial social welfare function is defined as a function for which there exists a single voter whose strict preferences are followed. This definition allows for many different dictatorial functions, since nonstrict preferences of the dictator are not necessarily followed. In particular, we construct examples of dictatorial functions which do not satisfy Transitivity and IIA. Thus Arrow’s theorem, in the case of nonstrict preferences, does not provide a complete characterization of all social welfare functions satisfying Transitivity, the Weak Pareto Principle, and IIA. The main results of this article provide such a characterization for Arrow’s theorem, as well as for follow up results by Wilson. In particular, we strengthen Arrow’s and Wilson’s result by giving an exact if and only if condition for a function to satisfy Transitivity and IIA (and the Weak Pareto Principle). Additionally, we derive formulae for the number of functions satisfying these conditions. 1
Optimal Inapproximability Results for MAXCUT and . . .
, 2005
"... In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of αGW + ɛ, for all ɛ> 0; here αGW ≈.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games Conjectur ..."
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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 αGW + ɛ, for all ɛ> 0; here αGW ≈.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games Conjecture of Khot [36] holds then the GoemansWilliamson approximation algorithm is optimal. Our result indicates that the geometric nature of the GoemansWilliamson algorithm might be intrinsic to the MAXCUT 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 [42]. A stronger version of this conjecture called Plurality Is Stablest is still open, although [42] contains a proof of an asymptotic version of it. Our techniques extend to several other twovariable constraint satisfaction problems. In particular, subject to the Unique Games Conjecture, we show tight or nearly tight hardness results for MAX2SAT, MAXqCUT, and MAX2LIN(q). For MAX2SAT we show approximation hardness up to a factor of roughly.943. This nearly matches the.940 approximation algorithm of Lewin, Livnat, and Zwick [40]. Furthermore, we show that our.943... factor is actually tight for a slightly restricted version of MAX2SAT. For MAXqCUT we show