Geometric optimization and unique sink orientations of cubes

Abstract

There is an ongoing attempt of designing a provably fast method for solving Linear Programs and related geometric optimization problems by combinatorial methods in the unit cost model (as opposed to the bit-model, where polynomial methods are known). Among others, this research has brought forth randomized methods with subexponential running time, but also a number of interesting combinatorial models like Oriented Matroids, Abstract Objective Functions (AOF), Unimodal Numberings, LP-type Problems, Abstract Optimization Problems (AOP), Unique Sink Orientations of Cubes (USO), et cetera. Although many of these models are quite general, so far there has been little success in showing good lower bounds even in these generalized abstractions. As for the simplex method, there are a number of open questions concerning several pivoting rules, notably randomized ones. After a general introduction, we will focus on one particular model: unique sink orientations of cubes (USO). To this end consider (the edge graph of) an n-dimensional hypercube with its edges oriented so that every face has a unique sink. Such an orientation is called a unique sink orientation, and we are interested in finding the unique sink of the whole cube when the orientation is given implicitly. The basic operation available is the so-called vertex evaluation, where we can access an arbitrary vertex of the cube, for which we obtain the orientations of the incident edges. Unique sink orientations occur when the edges of a deformed geometric n-dimensional cube (i.e., a polytope with the combinatorial structure of a cube) are oriented according to some generic linear function. These orientations are easily seen to be acyclic. The main motivation for studying unique sink orientations are certain linear complementarity problems, which allow this combinatorial abstraction (due to Alan Stickney and Layne Watson), where orientations with cycles can arise. Similarly, linear programming and some quadratic optimization problems, like computing the smallest enclosing ball of a finite point set, are polynomial time reducible (in the unit cost model) to finding a sink in a unique sink orientation (possibly with cycles). The talk surveys some results concerning upper and lower bounds for algorithms finding the sink in a USO (acyclic or possibly with cycles). © Springer-Verlag 2004.