On Geometric Structure of Global Roundings for Graphs and Range Spaces

Given a hypergraph H = (V,F) and a [0, 1]-valued vector a ∈ [0, 1] , its global rounding is a binary (i.e.,{0, 1}-valued) vector α ∈ {0, 1} such that | ∑ v∈F (a(v)−α(v))| < 1 holds for each F ∈ 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.