Optimizing some constructions with bars: new geometric knapsack problems

A set of vertical bars planted on given points of a horizontal line defines a fence composed of the quadrilaterals bounded by successive bars. A set of bars in the plane, each having one endpoint at the origin, defines an umbrella composed of the triangles bounded by successive bars. Given a collection of bars, we study how to use them to build the fence or the umbrella of maximum total area. We present optimal algorithms for these constructions. The problems introduced in this paper are related to the Geometric Knapsack problems [Arkin et al., Algorithmica 1993] and the Rearrangement Inequality [Wayne, Scripta Math. 1946].