In an undirected graph, a proper $(k,i)$-coloring is an assignment of a set of colors to each vertex such that any two adjacent vertices have at most common colors. The $(k,i)$-coloring problem is to compute the minimum number of colors required for a proper $(k,i)$-coloring. This is a generalization of the classical graph coloring problem. We design a parameterized algorithm for the $(k,i)$-coloring problem with the size of the feedback vertex set as a parameter. Our algorithm does not use tree-width machinery, thus answering a question of Majumdar, Neogi, Raman and Tale [CALDAM 2017]. We also give a faster exact algorithm for $(k,k-1)$-coloring. From the hardness perspective, we show that the $(k,i)$-coloring problem is NP-complete for any fixed values $i$, $k$, whenever $i < k$, thereby settling a conjecture of Méndez-Díaz and Zabala (1999) and again asked by Majumdar, Neogi, Raman and Tale. The NP-completeness result improves the partial NP-completeness shown in the preliminary version of this paper published in CALDAM 2018.