Spring 2020

On the Relationship Between Lattice Problems Across Different L_p Norms

Monday, Jun. 14, 2021 11:30 am12:00 pm PDT

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Eisenbrand and Venzin recently showed that the fastest algorithm for (large) constant-factor-approximate SVP and CVP in the L_2 norm can be made to work for all L_p norms. Their techniques are simple and elegant---the kind of proof that's surprising at first and then obvious in hindsight.

In follow-up work with Aggarwal, Chen, Kumar, and Li, we use their techniques to produce generic reductions, showing that any algorithm for SVP or CVP in one L_p norm yields algorithms in different L_p norms with only a constant factor loss in the approximation factor and a 2^{eps n} factor loss in the running time. (The reduction moves from larger p to smaller p for SVP and in the opposite direction for CVP.)

In my eyes (and perhaps only my eyes) all of this yields two surprising conclusions:
1) Lattice problems in different L_p norms are far more closely related than I originally thought. 
2) If we accept the complexity-theoretic assumptions behind the recent fine-grained lower bounds for CVP in L_p norms (with Aggarwal, Bennett, and Golovnev), then the time-approximation factor tradeoff for CVP in L_p norms must behave... strangely.

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