On geometric structure of global roundings for graphs and range spaces

Abstract

Given a hypergraph H = (V, ℱ) and a [0, 1]-valued vector a ∈ [0,1]V, its global rounding is a binary (i.e.,{0, 1}-valued) vector α ∈ {0,1}V such that |∑υ∈F (a(υ)-α(υ))| < 1 holds fo each F ε ℱ. We study geometric (or combinatorial) structure of the set of global roundings of a using the notion of compatible set with respect to the discrepancy distance. We conjecture that the set of global roundings forms a simplex if the hypergraph satisfies "shortest-path" axioms, and prove it for some special cases including some geometric range spaces and the shortest path hypergraph of a series-parallel graph. © Springer-Verlag Berlin Heidelberg 2004.