Coloring Random Non-Uniform Bipartite Hypergraphs

TL;DR

This paper introduces a polynomial-time algorithm for 2-coloring random non-uniform bipartite hypergraphs with high probability, given sufficient edge density, and discusses extending the approach to k-colorings.

Contribution

The paper presents a novel polynomial-time algorithm for 2-coloring non-uniform hypergraphs and analyzes its effectiveness under certain probabilistic conditions.

Findings

Successful 2-coloring with high probability when expected edges are at least dn ln n

Algorithm operates in polynomial time

Discussion on extending to k-coloring hypergraphs

Abstract

Let $H_{n,(p_m)_{m=2,\ldots,M}}$ be a random non-uniform hypergraph of dimension $M$ on $2n$ vertices, where the vertices are split into two disjoint sets of size $n$, and colored by two distinct colors. Each non-monochromatic edge of size $m=2,\ldots,M$ is independently added with probability $p_m$. We show that if $p_2,\ldots,p_M$ are such that the expected number of edges in the hypergraph is at least $dn\ln n$, for some $d>0$ sufficiently large, then with probability $(1-o(1))$, one can find a proper 2-coloring of $H_{n,(p_m)_{m=2,\ldots,M}}$ in polynomial time. We present a polynomial time algorithm for hypergraph 2-coloring, and provide discussions on extension of the approach for $k$-coloring of non-uniform hypergraphs.