Structured $(\min,+)$-Convolution And Its Applications For The Shortest Vector, Closest Vector, and Separable Nonlinear Knapsack Problems

TL;DR

This paper introduces efficient algorithms for $( ext{min},+)$-convolution in various complex cases, enabling improved solutions for knapsack and lattice problems with nonlinear objectives.

Contribution

It develops the first sub-quadratic algorithms for convex, concave, piece-wise linear, and polynomial cases of $( ext{min},+)$-convolution, expanding the problem's applicability.

Findings

Developed sub-quadratic algorithms for new convolution cases.

Applied algorithms to knapsack and lattice problems.

Enhanced computational efficiency for nonlinear objective functions.

Abstract

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 $f(x) =\beta + \alpha \cdot x$; 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.