Balanced Independent and Dominating Sets on Colored Interval Graphs

We study two new versions of independent and dominating set problems on vertex-colored interval graphs, namely $f$-Balanced Independent Set ($f$-BIS) and $f$-Balanced Dominating Set ($f$-BDS). Let $G=(V,E)$ be a vertex-colored interval graph with a color assignment function $\gamma \colon V \rightarrow \{1,\ldots,k\}$ that maps all vertices in $G$ onto $k$ colors. A subset of vertices $S\subseteq V$ is called $f$-balanced if $S$ contains $f$ vertices from each color class. In the $f$-BIS and $f$-BDS problems, the objective is to compute an independent set or a dominating set that is $f$-balanced. We show that both problems are NP-complete even on proper interval graphs. For the $f$-BIS problem, we design two FPT algorithms, one parameterized by $(f,k)$ for interval graphs and the other parameterized by the vertex cover number for general graphs. Moreover, for an optimization variant of BIS on interval graphs, we show that a simple greedy approach achieves an approximation ratio of $2$.