Quasi-polynomial Algorithms for List-coloring of Nearly Intersecting Hypergraphs

A hypergraph $\mathcal{H}$ on $n$ vertices and $m$ edges is said to be {\it nearly-intersecting} if every edge of $\mathcal{H}$ intersects all but at most polylogarthmically many (in $m$ and $n$) other edges. Given lists of colors $\mathcal{L}(v)$, for each vertex $v\in V$, $\mathcal{H}$ is said to be $\mathcal{L}$-(list) colorable, if each vertex can be assigned a color from its list such that no edge in $\mathcal{H}$ is monochromatic. We show that list-colorability for any nearly intersecting hypergraph, and lists drawn from a set of constant size, can be checked in quasi-polynomial time in $m$ and $n$.

💡 Research Summary

The paper studies the list‑coloring problem on hypergraphs that are “nearly‑intersecting”: every hyperedge intersects all but at most a polylogarithmic number of other edges (in the parameters (m) and (n)). Formally, a hypergraph (\mathcal H=(V,\mathcal E)) is (c)-intersecting if each edge fails to intersect at most (c) other edges; when (c=\operatorname{polylog}(m,n)) the hypergraph is called nearly‑intersecting. Each vertex (v) is given a list (L(v)\subseteq