Balanced Coloring: Equally Easy for All Numbers of Colors?

Abstract

We investigate the problem to color the vertex set of a hypergraph H = (X, ε) with a fixed number of colors in a balanced manner, i.e., in such a way that all hyperedges contain roughly the same number of vertices in each color (discrepancy problem). We show the following result:

Suppose that we are able to compute for each induced subhypergraph a coloring in c₁ colors having discrepancy at most D. Then there are colorings in arbitrary numbers c₂ of colors having discrepancy at most 11/10 c ²₁ D. A c₂-coloring having discrepancy at most 11/10 c ²₁ D+3c₁ ^-k∣X∣ can be computed from (c₁-1)(c₂-1)k colorings in c₁ colors having discrepancy at most D with respect to a suitable subhypergraph of H. A central step in the proof is to show that a fairly general rounding problem (linear discrepancy problem in c₂ colors) can be solved by computing low-discrepancy c₁-colorings.