On the power of discrete and of lexicographic Helly-type theorems

Helly´s theorem says that if every d + 1 elements of a given finite set of convex objects in R^d have a common point, then there is a point common to all of the objects in the set. We define three 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 Helly theorems and their corresponding (standard) Helly theorems. We obtain several discrete and lexicographic Helly numbers. Using these types of Helly theorems we get linear time solutions for various optimization problems. For this, we define a framework, DLP-type (discrete linear programming type), and provide algorithms that solve in randomized linear time fixed-dimensional DLP-type problems. We show that the complexity of the DLP-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 p-center on the real line, the discrete weighted 1-center problem in R^d with l_∞ norm, the standard (continuous) problem of finding a line transversal for a totally separable set of planar convex objects, a discrete version of the problem of finding a line transversal for a set of axis-parallel planar 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.