Three Candidate Plurality is Stablest for Correlations at most 1/10
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We prove the three candidate Plurality is Stablest Conjecture of Khot-Kindler-Mossel-O'Donnell from 2005 for correlations ρ satisfying -1/43<ρ<1/10: the Plurality function is the most noise stable three candidate election method with small influences, when the corrupted votes have correlation -1/43<ρ<1/10 with the original votes. The previous best result of this type only achieved positive correlations at most 10^-10^10. Our result follows by solving the three set Standard Simplex Conjecture of Isaksson-Mossel from 2011 for all correlations -1/43<ρ<1/10. The Gaussian Double Bubble Theorem corresponds to the case ρ→1^-, so in some sense, our result is a generalization of the Gaussian Double Bubble Theorem. Our result is also notable since it is the first result for any ρ<0, which is the only relevant case for computational hardness of MAX-3-CUT. In fact, assuming the Unique Games Conjecture, we show that MAX-3-CUT is NP-hard to approximate within a multiplicative factor of .98937, which improves on the known (unconditional) NP-hardness of approximation within a factor of 1-(1/102), proven in 1997. As an additional corollary, we conclude that three candidate Borda Count is stablest for all -1/43<ρ<1/10.
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