Approximate Hypergraph Coloring under Low-discrepancy and Related Promises

A hypergraph is said to be \(\chi\)-colorable if its vertices can be colored with \(\chi\) 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 \(\approx n^{1-1/k}\) 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 \(\ell \ll \sqrt{k}\), we give an algorithm to color the it with \(\approx n^{O(\ell^2/k)}\) 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 \(\ell = O(\log k)\) (resp. \(\ell=2\)). 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 \((k-\ell)\)-coloring such that each hyperedge is polychromatic with all these colors, we give a \(2\)-coloring algorithm that miscolors at most \(k^{-\Omega(k)}\) of the hyperedges when \(\ell \ll \sqrt{k}\), and complement this with a matching UG hardness result showing that when \(\ell =\sqrt{k}\), it is hard to even beat the \(2^{-k+1}\) bound achieved by a random coloring.