On the Complexity of Solving or Approximating Convex Recoloring Problems

Given a graph with an arbitrary vertex coloring, the Convex Recoloring Problem (CR) consists of recoloring the minimum number of vertices so that each color induces a connected subgraph. We focus on the complexity and inapproximabiliy of this problem on k-colored graphs, for fixed k ≥ 2. We prove a very strong complexity result showing that CR is already NP-hard on k-colored grids, and therefore also on planar graphs with maximum degree 4. For each k ≥ 2, we also prove that, for a positive constant c, there is no cln n-approximation algorithm even for k-colored n-vertex bipartite graphs, unless P = NP. For 2-colored (q,q − 4)-graphs, a class that includes cographs and P 4-sparse graphs, we present polynomial-time algorithms for fixed q. The same complexity results are obtained for a relaxation of CR, where only one fixed color is required to induce a connected subgraph.