Hardness and inapproximability of convex recoloring problems

Given a graph with an arbitrary vertex coloring, the Convex Recoloring Problem (CR) consists in recoloring the minimum number of vertices so that each color induces a connected subgraph. We focus on the complexity and inapproximability of this problem on k  -colored graphs, for fixed k⩾2k⩾2. We prove a strong complexity result showing that, for each k⩾2k⩾2, CR is already NP-hard on k  -colored grids, and therefore also on planar graphs with maximum degree 4. For each k⩾2k⩾2, we prove that, for a positive constant c  , there is no clnn-approximation algorithm for k-colored n  -vertex (bipartite) graphs, unless P=NPP=NP. We also prove that CR parameterized by the number of color changes is W[2]W[2]-hard. For 2-colored (q,q−4)(q,q−4)-graphs, a class that includes cographs and P4P4-sparse graphs, we present linear-time algorithms for fixed q. The same complexity and inapproximability results are obtained for two relaxations of the problem, where only one fixed color or any color is required to induce a connected subgraph, respectively.