A Fast Coloring Oracle for Average Case Hypergraphs

Hypergraph $2$-colorability is one of the classical NP-hard problems. Person and Schacht [SODA'09] designed a deterministic algorithm whose expected running time is polynomial over a uniformly chosen $2$-colorable $3$-uniform hypergraph. Lee, Molla, and Nagle recently extended this to $k$-uniform hypergraphs for all $k\geq 3$. Both papers relied heavily on the regularity lemma, hence their analysis was involved and their running time hid tower-type constants. Our first result in this paper is a new simple and elementary deterministic $2$-coloring algorithm that reproves the theorems of Person-Schacht and Lee-Molla-Nagle while avoiding the use of the regularity lemma. We also show how to turn our new algorithm into a randomized one with average expected running time of only $O(n)$. Our second and main result gives what we consider to be the ultimate evidence of just how easy it is to find a $2$-coloring of an average $2$-colorable hypergraph. We define a coloring oracle to be an algorithm which, given vertex $v$, assigns color red/blue to $v$ while inspecting as few edges as possible, so that the answers to any sequence of queries to the oracle are consistent with a single legal $2$-coloring of the input. Surprisingly, we show that there is a coloring oracle that, on average, can answer every vertex query in time $O(1)$.