结构化$(\min,+)$ -卷积及其在最短向量、最近向量和可分非线性背包问题中的应用
Structured $(\min,+)$-Convolution And Its Applications For The Shortest Vector, Closest Vector, and Separable Nonlinear Knapsack Problems
摘要：
在这项工作中，我们考虑的问题，计算的$(\min, +)$卷积的两个序列$a$和$b$的长度$n$和$m$，分别，其中$n \geq m$。我们假设$a$是任意的，但是$b_i = f(i)$，其中$f(x) \colon [0,m) \to \mathbb{R}$是具有以下性质之一的函数：1.线性情况下，当$f(x) =β+ α\cdot x$ ; 2.单调的情况下，当$f(i+1) \geq f(i)$，对于任何$i$ ; 3.凸的情况下，当$f(i+1) - f(i) \geq f(i) - f(i-1)$，对于任何$i$ ; 4.当为$f(i+1) - f(i) \leq f(i) - f(i-1)$时，任何$i$的凹面情况; 5.分段线性情况，当$f(x)$由$p$线性片段组成时; 6.多项式的情况下，当$f \in \mathbb{Z}^d[x]$，对于一些固定的$d$。据我们所知，病例4-6在以前的文献中没有被考虑过。我们为他们开发真正的次二次算法。我们将我们的结果应用到背包问题与可分离的非线性目标函数，最短的格向量，最接近的格向量的问题。
英文摘要：
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 $n \geq m$. We assume that $a$ is arbitrary, but $b_i = f(i)$, where $f(x) \colon [0,m) \to \mathbb{R}$ is a function with one of the following properties: 1. the linear case, when $f(x) =β+ α\cdot x$; 2. the monotone case, when $f(i+1) \geq f(i)$, for any $i$; 3. the convex case, when $f(i+1) - f(i) \geq f(i) - f(i-1)$, for any $i$; 4. the concave case, when $f(i+1) - f(i) \leq f(i) - f(i-1)$, for any $i$; 5. the piece-wise linear case, when $f(x)$ consist of $p$ linear pieces; 6. the polynomial case, when $f \in \mathbb{Z}^d[x]$, 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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