Structured (min,+)-Convolution And Its Applications For The Shortest Vector, Closest Vector, and Separable Nonlinear Knapsack Problems
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In this work we consider the problem of computing the (min, +)-convolution of two sequences a and b of lengths n and m, respectively, where n ≥ m. We assume that a is arbitrary, but b_i = f(i), where f(x) [0,m) →ℝ is a function with one of the following properties: 1. the linear case, when f(x) =β + α· x; 2. the monotone case, when f(i+1) ≥ f(i), for any i; 3. the convex case, when f(i+1) - f(i) ≥ f(i) - f(i-1), for any i; 4. the concave case, when f(i+1) - f(i) ≤ f(i) - f(i-1), for any i; 5. the piece-wise linear case, when f(x) consist of p linear pieces; 6. the polynomial case, when f ∈ℤ^d[x], for some fixed d. To the best of our knowledge, the cases 4-6 were not considered in literature before. We develop true sub-quadratic algorithms for them. We apply our results to the knapsack problem with a separable nonlinear objective function, shortest lattice vector, and closest lattice vector problems.
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