Approximate Hypergraph Coloring under Low-discrepancy and Related Promises

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 trivially 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: • Low-discrepancy: If the hypergraph has a 2-coloring of discrepancy l≪ √ k, we give an algorithm to color the hypergraph with ≈ nO(l/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 Unique Games conjecture, we improve the latter hardness factor to 2−O(k) for almost discrepancy-1 hypergraphs. • Rainbow colorability: If the hypergraph has a (k− l)-coloring such that each hyperedge is polychromatic with all these colors (this is stronger than a (l+ 1)-discrepancy 2-coloring), 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 Unique Games hardness result showing that when l = √ k, it is hard to even beat the 2−k+1 bound achieved by a random coloring. ∗Supported by NSF CCF-1115525 [email protected] †Supported in part by NSF grant CCF-1115525. [email protected] ‡Supported by a Samsung Fellowship and NSF CCF-1115525. [email protected]