The Polynomial-Time Low-Degree Conjecture
Prior state unknown→disproved
the construction is probabilistic; an explicit uniformly samplable example remains open
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theoretical-computer-science / Average-case complexity
The Polynomial-Time Low-Degree Conjecture
The low-degree conjecture predicts that when the low-degree advantage between a planted distribution and a uniform null distribution stays bounded, no polynomial-time algorithm can distinguish them. It is false. There is a planted distribution that agrees with the null through the relevant degree, is invariant under vertex relabeling, and is nevertheless distinguished in polynomial time by a rank argument.
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The low-degree conjecture predicts that when the low-degree advantage between a planted distribution and a uniform null distribution stays bounded, no polynomial-time algorithm can distinguish them. It is false. There is a planted distribution that agrees with the null through the relevant degree, is invariant under vertex relabeling, and is nevertheless distinguished in polynomial time by a rank argument.
the construction is probabilistic; an explicit uniformly samplable example remains open
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