Minkowski Sum Selection and Finding

TL;DR

This paper presents optimal algorithms for Minkowski Sum Selection and Finding problems with linear and specific nonlinear objectives, improving computational efficiency for fixed parameters.

Contribution

It introduces optimal time algorithms for Minkowski Sum Selection and Finding problems, including deterministic and randomized approaches for fixed parameters.

Findings

Optimal $O(n ext{log} n)$ algorithms for $ ext{lambda}=1$

Deterministic $O(n ext{log}^2 n)$ and randomized $O(n ext{log} n)$ algorithms for fixed $ ext{lambda}>1$

Optimal $O(n ext{log} n)$ algorithms for Minkowski Sum Finding with specific objectives

Abstract

For the \textsc{Minkowski Sum Selection} problem with linear objective functions, we obtain the following results: (1) optimal $O(n\log n)$ time algorithms for $\lambda=1$; (2) $O(n\log^2 n)$ time deterministic algorithms and expected $O(n\log n)$ time randomized algorithms for any fixed $\lambda>1$. For the \textsc{Minkowski Sum Finding} problem with linear objective functions or objective functions of the form $f(x,y)=\frac{by}{ax}$, we construct optimal $O(n\log n)$ time algorithms for any fixed $\lambda\geq 1$.