Discrete and Lexicographic Helly Theorems and their Relations to LP-Type Problems

Helly’s theorem says that if every d + 1 elements of a given finite set of convex objects in IR have a common point, then there is a point common to all of the objects in the set. We define three new types of Helly theorems: discrete Helly theorems where the common point should belong to an a-priori given set, lexicographic Helly theorems where the common point should not be lexicographically greater than a given point, and lexicographic-discrete Helly theorems. We show the relations between these new Helly theorems and their corresponding (standard) Helly theorems. We obtain several new discrete and lexicographic Helly numbers. Using these new types of Helly theorems we get linear time solutions for various optimization problems. For this, we define a new framework, discrete LP-type (LP stands for Linear Programming), and provide new algorithms that solve in randomized linear time fixed-dimensional discrete LP-type problems. We show that the complexity of the discrete LP-type class stands somewhere between Linear Programming (LP) and Integer Programming (IP). Finally, we use our results in order to solve in randomized linear time problems such as the discrete pcenter on the real line, the discrete weighted 1-center problem in IR with l∞ norm, the standard (continuous) and a discrete version of the optimization problem of finding a line transversal for a finite set of planar axisparallel rectangles, and the (planar) lexicographic rectilinear p-center problem for p = 1, 2, 3. These are the first known linear time algorithms for these problems.