Proceedings
Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

Improved Inapproximability of Rainbow Coloring

Abstract

A rainbow q-coloring of a k-uniform hypergraph is a q-coloring of the vertex set such that every hyperedge contains all q colors.
We prove that given a rainbow -colorable k-uniform hypergraph, it is NP-hard to find a normal 2-coloring. Previously, this was only known for rainbow -colorable hypergraphs (Guruswami and Lee, SODA 2015).
We also study a generalization which we call rainbow (q, p)-coloring, defined as a coloring using q colors such that every hyperedge contains at least p colors. We prove that given a rainbow -colorable k uniform hypergraph, it is NP-hard to find a normal c-coloring for any c = o(k).
The proof of our second result relies on two combinatorial theorems. One of the theorems was proved by Sarkaria (J. Comb. Theory, Ser. B 1990) using topological methods and the other theorem we prove using a generalized Borsuk-Ulam theorem.

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cover image Proceedings
Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms
Pages: 1479 - 1495
Editor: Shuchi Chawla
ISBN (Online): 978-1-611975-99-4

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Published online: 23 December 2019

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