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.
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Doerr, B. (2002). Balanced Coloring: Equally Easy for All Numbers of Colors?. In: Alt, H., Ferreira, A. (eds) STACS 2002. STACS 2002. Lecture Notes in Computer Science, vol 2285. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45841-7_8
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DOI: https://doi.org/10.1007/3-540-45841-7_8
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