A Note on the Concrete Hardness of the Shortest Independent Vectors Problem in Lattices
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BlΓΆmer and Seifert showed that π²π¨π΅π―_2 is NP-hard to approximate by giving a reduction from π’π΅π―_2 to π²π¨π΅π―_2 for constant approximation factors as long as the π’π΅π― instance has a certain property. In order to formally define this requirement on the π’π΅π― instance, we introduce a new computational problem called the Gap Closest Vector Problem with Bounded Minima. We adapt the proof of BlΓΆmer and Seifert to show a reduction from the Gap Closest Vector Problem with Bounded Minima to π²π¨π΅π― for any β_p norm for some constant approximation factor greater than 1. In a recent result, Bennett, Golovnev and Stephens-Davidowitz showed that under Gap-ETH, there is no 2^o(n)-time algorithm for approximating π’π΅π―_p up to some constant factor Ξ³β₯ 1 for any 1 β€ p β€β. We observe that the reduction in their paper can be viewed as a reduction from π¦πΊπ3π²π π³ to the Gap Closest Vector Problem with Bounded Minima. This, together with the above mentioned reduction, implies that, under Gap-ETH, there is no 2^o(n)-time algorithm for approximating π²π¨π΅π―_p up to some constant factor Ξ³β₯ 1 for any 1 β€ p β€β.
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