On the Algorithmic Aspects of Discrete and Lexicographic Helly-Type Theorems and the Discrete LP-Type Model

Abstract

Helly's theorem says that, if every $d+1$ elements of a given finite set of convex objects in $\mathbb{R}^d$ have a common point, there is a point common to all of the objects in the set. In discrete Helly theorems the common point should belong to an a priori given set. In lexicographic Helly theorems the common point should not be lexicographically greater than a given point. Using discrete and lexicographic Helly theorems we get linear time solutions for various optimization problems. For this, we introduce the DLP-type (discrete linear programming-type) model, and provide new algorithms that solve in randomized linear time fixed-dimensional DLP-type problems. For variable-dimensional DLP-type problems, our algorithms run in time subexponential in the combinatorial dimension. 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 $\mathbb{R}^d$ with either $l_1$ or $l_\infty$ 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. Moreover, we use our algorithms to solve in randomized subexponential time various problems in game theory, improving upon the best known algorithms for these problems.