Minkowski Sum Selection and Finding

Abstract

Let P,Q ⊆ ℝ² be two n-point multisets and Ar ≥ b be a set of λ inequalities on x and y, where A ∈ ℝ^λ×2, $r=[^x_y]$, and b ∈ ℝ^λ. Define the constrained Minkowski sum (P ⊕ Q) _Ar ≥ b as the multiset {(p + q) | p ∈ P,q ∈ Q,A(p + q) ≥ b}. Given P, Q, Ar ≥ b, an objective function f:ℝ²→ℝ, and a positive integer k, the Minkowski Sum Selection problem is to find the k ^th largest objective value among all objective values of points in (P ⊕ Q) _Ar ≥ b . Given P, Q, Ar ≥ b, an objective function f:ℝ²→ℝ, and a real number δ, the Minkowski Sum Finding problem is to find a point (x ^*,y ^*) in (P ⊕ Q) _Ar ≥ b such that |f(x ^*,y ^*) − δ| is minimized. For the Minkowski Sum Selection problem with linear objective functions, we obtain the following results: (1) optimal O(nlogn) time algorithms for λ= 1; (2) O(nlog² n) time deterministic algorithms and expected O(nlogn) time randomized algorithms for any fixed λ> 1. For the Minkowski Sum Finding problem with linear objective functions or objective functions of the form $f(x,y)=\frac{by}{ax}$, we construct optimal O(nlogn) time algorithms for any fixed λ ≥ 1. As a byproduct, we obtain improved algorithms for the Length-Constrained Sum Selection problem and the Density Finding problem.