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 an interval graph with a color assignment function γ: V → {1,…,k} that maps all vertices in G onto k colors. A subset of vertices S ⊆ 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 BIS problem on interval graphs, we design two FPT algorithms, one parameterized by (f, k) and the other by the vertex cover number of G. Moreover, for an optimization variant of BIS on interval graphs, we present a polynomial time approximation scheme (PTAS) and an O(n log n) time 2-approximation algorithm.
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