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Discrete Math

On the hardness of coloring rainbow-colorable hypergraphs

Aditya Potukuchi, Rutgers

Location:  Hill 705
Date & time: Monday, 02 April 2018 at 2:00PM - 3:00PM

Abstract:  A (uniform) hypergraph is c-colorable if its vertices can be assigned colors from 1,...,c so that no hyperedge is monochromatic. A hypergraph is r-rainbow colorable (or r-polychromatic) if its vertices can be assigned colors from 1,...,r so that every edge has all r colors. It is very easy to see that if a hypergraph is r-rainbow colorable for r geq 2, then it is 2-colorable. However, given a hypergraph with the promise that is r-rainbow colorable for r geq 2, finding an efficient algorithm that outputs a proper 2-coloring of it does not always seem to be an easy problem. This result shows that finding a c-coloring of a k-uniform, k(1 - o_c(1))-rainbow colorable hypergraph is NP-hard. The gadgets involved in the reduction are some kinds of variations/generalizations of the Kneser hypergraph, some of which seem to be well studied, and some which are not (as far as we know).

This is based on joint work with Per Austrin and Amey Bhangale.

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