Color-bounded hypergraphs, VI: Structural and functional jumps in complexity

Abstract

A stably bounded hypergraph $\mathcal{H}$ is a hypergraph together with four color-bound functions $s$, $t$, $a$ and $b$, each assigning positive integers to the edges. A vertex coloring of $\mathcal{H}$ is considered proper if each edge $E$ has at least $s\left(E\right)$ and at most $t\left(E\right)$ different colors assigned to its vertices, moreover each color occurs on at most $b\left(E\right)$ vertices of $E$, and there exists a color which is repeated at least $a\left(E\right)$ times inside $E$. The lower and the upper chromatic number of $\mathcal{H}$ is the minimum and the maximum possible number of colors, respectively, over all proper colorings. An interval hypergraph is a hypergraph whose vertex set allows a linear ordering such that each edge is a set of consecutive vertices in this order.

We study the time complexity of testing colorability and determining the lower and upper chromatic numbers. A complete solution is presented for interval hypergraphs without overlapping edges. Complexity depends both on problem type and on the combination of color-bound functions applied, except that all the three coloring problems are NP-hard for the function pair $a,b$ and its extensions. For the tractable classes, linear-time algorithms are designed. It also depends on problem type and function set whether complexity jumps from polynomial to NP-hard if the instance is allowed to contain overlapping intervals. Comparison is facilitated with three handy tables which also include further structure classes.