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 $\lambda\geq 1$.
Categories: cs.DS cs.CG