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...

The minrank of a graph G is the minimum rank of a matrix M that can be obtained from the adjacency matrix of G by switching some ones to zeros (i.e., deleting edges) and then setting all diagonal entries to one. This quantity is closely related to the fundamental information-theoretic problems of (linear) index coding (Bar-Yossef et al., FOCS'06), network coding and distributed storage, and to Valiant's approach for proving superlinear circuit lower bounds (Valiant, Boolean Function Complexity '92).We prove tight bounds on the minrank of random Erdős-Rényi graphs G(n, p) for all regimes of p ∈ [0, 1]. In particular, for any constant p, we show that minrk(G) = Θ(n/ log n) with high probability, where G is chosen from G(n, p). This bound gives a near quadratic improvement over the previous best lower bound of Ω( √ n) (Haviv and Langberg, ISIT'12), and partially settles an open problem raised by Lubetzky and Stav (FOCS '07). Our lower bound matches the well-known upper bound obtained by the "clique covering" solution, and settles the linear index coding problem for random graphs. Finally, our result suggests a new avenue of attack, via derandomization, on Valiant's approach for proving superlinear lower bounds for logarithmic-depth semilinear circuits.

We show that static data structure lower bounds in the group (linear) model imply semiexplicit lower bounds on matrix rigidity. In particular, we prove that an explicit lower bound of t ≥ ω(log 2 n) on the cell-probe complexity of linear data structures in the group model, even against arbitrarily small linear space (s = (1 + ε)n), would already imply a semi-explicit (P NP ) construction of rigid matrices with significantly better parameters than the current state of art (Alon, Panigrahy and Yekhanin, 2009). Our results further assert that polynomial (t ≥ n δ ) data structure lower bounds against near-optimal space, would imply super-linear circuit lower bounds for log-depth linear circuits (a four-decade open question). In the succinct space regime (s = n + o(n)), we show that any improvement on current cell-probe lower bounds in the linear model would also imply new rigidity bounds. Our results rely on a new connection between the "inner" and "outer" dimensions of a matrix (Paturi and Pudlák, 2006), and on a new reduction from worst-case to average-case rigidity, which is of independent interest.

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