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  • Miklós Ajtai
  • 1996
We give a random class of lattices in Z n so that, if there is a probabilistic polynomial time algorithm which nds a short vector in a random lattice with a probability of at least 1 2 then there is also a probabilistic polynomial time algorithm which solves the following three lattice problems in every lattice in Z n with a probability exponentially close(More)
Let G be a (k + I)-graph (a hypergraph with each hyperedge of size k + 1) with n vertices and average degree t. Assume k Q t Q n. If G is uncrowded (contains no cycle of size 2, 3, dr 4) then there exists an independent set of size c,(n/t)(ln t)'lk. Let G be a graph with n vertices and average valence t. Turan's theorem implies a(G) > n/(t + 1). (See(More)
We present a randomized 2^{<italic>O(n)</italic>} time algorithm to compute a shortest non-zero vector in an <italic>n</italic>-dimensional rational lattice. The best known time upper bound for this problem was 2^{<italic>O(n</italic>\log <italic>n</italic>)} first given by Kannan [7] in 1983. We obtain several consequences of this algorithm for related(More)