Minkowski Sum Selection and Finding

Let P,Q ⊆ R be two n-point multisets and Ar ≥ b be a set of λ inequalities on x and y, where A ∈ R, r = [y ], and b ∈ R. 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 : R → R, and a positive integer k, the Minkowski Sum Selection problem is to find the k largest objective value among all objective values of points in (P⊕Q)Ar≥b. Given P , Q, Ar ≥ b, an objective function f : R → 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(n logn) time algorithms for λ = 1; (2) O(n log n) time deterministic algorithms and expected O(n logn) 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) = by ax , we construct optimal O(n log n) 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.