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 classic 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, and make progress on the NP-completeness of specific cases of $(k,i)$-coloring.