For odd integers p ≥ 1 (and p = ∞), we show that the Closest Vector Problem in the p norm (CVP p ) over rank n lattices cannot be solved in 2 (1−ε)n time for any constant ε > 0 unless the Strong Exponential Time Hypothesis (SETH) fails. We then extend this result to "almost all" values of p ≥ 1, not including the even integers. This comes tantalizingly close to settling the quantitative time complexity of the important special case of CVP 2 (i.e., CVP in the Euclidean norm), for which a 2 n+o(n) -time algorithm is known. In particular, our result applies for any p = p(n) = 2 that approaches 2 as n → ∞.We also show a similar SETH-hardness result for SVP ∞ ; hardness of approximating CVP p to within some constant factor under the so-called Gap-ETH assumption; and other quantitative hardness results for CVP p 1 WLTB11, Laa15, BDGL16], and there is some reason to believe that the provably correct [ADRS15] algorithm can be improved. In particular, there is a provably correct 2 n/2+o(n) -time algorithm that approximates SVP 2 up to a small constant approximation factor [ADRS15].A different line of work extended the randomized sieving approach of [AKS01] to obtain 2 O(n)time algorithms for SVP in additional norms. In particular, Blömer and Naewe extended it to all p norms [BN09]. Subsequent work extended this further, first to arbitrary symmetric norms [AJ08] and then to the "near-symmetric norms" that arise in integer programming [Dad12].Finally, a third line of work extended the [AKS01] approach to approximate CVP. Ajtai, Kumar, and Sivakumar themselves showed a 2 O(n) -time algorithm for approximating CVP 2 to within any constant approximation factor strictly greater than one [AKS02]. Blömer and Naewe obtained the same result for all p norms [BN09], and Dadush extended it further to arbitrary symmetric norms and again to "near-symmetric norms" [Dad12]. We stress, however, that none of these results apply to exact CVP, and indeed, there are some barriers to extending these algorithms to exact CVP. (See, e.g., [ADS15].) Exact algorithms for CVP.Exact CVP appears to be a much more subtle problem than exact SVP. 2 Indeed, progress on exact CVP has been much slower than the progress on exact SVP. Over a decade after [AKS01], Micciancio and Voulgaris presented the first 2 O(n) -time algorithm for exact CVP 2 [MV13], using elegant new techniques built upon the approach of Sommer, Feder, and Shalvi [SFS09]. Specifically, they achieved a running time of 4 n+o(n) , and subsequent work even showed a running time of 2 n+o(n) for CVP 2 with Preprocessing (in which the algorithm is allowed access to arbitrary advice that depends on the lattice but not the target vector; see Section 2.1) [BD15]. Later, [ADS15] showed a 2 n+o(n) -time algorithm for CVP 2 , so that the current best known asymptotic running time is actually the same for SVP 2 and CVP 2 .However, for p = 2, progress for exact CVP p has been minimal. Indeed, the fastest known algorithms for exact CVP p with p = 2 are still the n O(n) -time enumeration algorithms first d...

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Let X = {f1, …, fn} be a set of scalar functions of the form fi : ℝ2 → ℝ which satisfy some natural properties. We describe a subdivision algorithm for computing a clustered ε‐isotopic approximation of the minimization diagram of X. By exploiting soft predicates and clustering of Voronoi vertices, our algorithm is the first that can handle arbitrary degeneracies in X, and allow scalar functions which are piecewise smooth, and not necessarily semi‐algebraic. We apply these ideas to the computation of anisotropic Voronoi diagram of polygonal sets; this is a natural generalization of anisotropic Voronoi diagrams of point sites, which extends multiplicatively weighted Voronoi diagrams. We implement a prototype of our anisotropic algorithm and provide experimental results.