A hypergraph is said to be χ-colorable if its vertices can be colored
with χ colors so that no hyperedge is monochromatic. 2-colorability is a
fundamental property (called Property B) of hypergraphs and is extensively
studied in combinatorics. Algorithmically, however, given a 2-colorable
k-uniform hypergraph, it is NP-hard to find a 2-coloring miscoloring fewer
than a fraction 2−k+1 of hyperedges (which is achieved by a random
2-coloring), and the best algorithms to color the hypergraph properly require
≈n1−1/k colors, approaching the trivial bound of n as k
increases.
In this work, we study the complexity of approximate hypergraph coloring, for
both the maximization (finding a 2-coloring with fewest miscolored edges) and
minimization (finding a proper coloring using fewest number of colors)
versions, when the input hypergraph is promised to have the following stronger
properties than 2-colorability:
(A) Low-discrepancy: If the hypergraph has discrepancy ℓ≪k,
we give an algorithm to color the it with ≈nO(ℓ2/k) colors.
However, for the maximization version, we prove NP-hardness of finding a
2-coloring miscoloring a smaller than 2−O(k) (resp. k−O(k))
fraction of the hyperedges when ℓ=O(logk) (resp. ℓ=2). Assuming
the UGC, we improve the latter hardness factor to 2−O(k) for almost
discrepancy-1 hypergraphs.
(B) Rainbow colorability: If the hypergraph has a (k−ℓ)-coloring such
that each hyperedge is polychromatic with all these colors, we give a
2-coloring algorithm that miscolors at most k−Ω(k) of the
hyperedges when ℓ≪k, and complement this with a matching UG
hardness result showing that when ℓ=k, it is hard to even beat the
2−k+1 bound achieved by a random coloring.Comment: Approx 201