Why we couldn't prove SETH hardness of the Closest Vector Problem for even norms, and of the Subset Sum Problem!
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Recent work [BGS17,ABGS19] has shown SETH hardness of some constant factor approximate CVP in the ℓ_p norm for any p that is not an even integer. This result was shown by giving a Karp reduction from k-SAT on n variables to approximate CVP on a lattice of rank n. In this work, we show a barrier towards proving a similar result for CVP in the ℓ_p norm where p is an even integer. We show that for any c, c'>0, if for every k > 0, there exists an efficient reduction that maps a k-SAT instance on n variables to a (1+exp(-n^c)))-CVP instance for a lattice of rank at most n^c' in the Euclidean norm, then 𝖼𝗈𝖭𝖯⊂𝖭𝖯/𝖯𝗈𝗅𝗒. We prove a similar result for (1+exp(-n^c)))-CVP for all even norms under a mild additional promise that the ratio of the distance of the target from the lattice and the shortest non-zero vector in the lattice is bounded by exp(n^O(1)). Furthermore, we show that for any c,c' > 0, and any even integer p, if for every k > 0, there exists an efficient reduction that maps a k-SAT instance on n variables to a (1+exp(-n^c)))-SVP_p instance for a lattice of rank at most n^c', then 𝖼𝗈𝖭𝖯⊂𝖭𝖯/𝖯𝗈𝗅𝗒. The result for SVP does not require any additional promise. While prior results have indicated that lattice problems in the ℓ_2 norm (Euclidean norm) are easier than lattice problems in other norms, this is the first result that shows a separation between these problems. We achieve this by using a result by Dell and van Melkebeek [JACM, 2014] on the impossibility of the existence of a reduction that compresses an arbitrary k-SAT instance into a string of length 𝒪(n^k-ϵ) for any ϵ>0. In addition to CVP, we also show that the same result holds for the Subset-Sum problem using similar techniques.
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