In an undirected graph, a proper $(k,i)$-coloring is an assignment of a set of $k$ colors to each vertex such that any two adjacent vertices have at most $i$ 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 show 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 and simpler 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 Mendez-Diaz 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.