On the complexity of restoring corrupted colorings

In the \(r\)-Fix problem, we are given a graph G, a (non-proper) vertex-coloring \(c : V(G) \rightarrow [r]\), and a positive integer k. The goal is to decide whether a proper r-coloring \(c'\) is obtainable from c by recoloring at most k vertices of G. Recently, Junosza-Szaniawski et al. (in: SOFSEM 2015: theory and practice of computer science, Springer, Berlin, 2015) asked whether the problem has a polynomial kernel parameterized by the number of recolorings k. In a full version of the manuscript, the authors together with Garnero and Montealegre, answered the question in the negative: for every \(r \ge 3\), the problem \(r\)-Fix does not admit a polynomial kernel unless . Independently of their work, we give an alternative proof of the theorem. Furthermore, we study the complexity of \(r\)-Swap, where the only difference from \(r\)-Fix is that instead of k recolorings we have a budget of k color swaps. We show that for every \(r \ge 3\), the problem \(r\)-Swap is -hard whereas \(r\)-Fix is known to be FPT. Moreover, when r is part of the input, we observe both Fix and Swap are -hard parameterized by the treewidth of the input graph. We also study promise variants of the problems, where we are guaranteed that a proper r-coloring \(c'\) is indeed obtainable from c by some finite number of swaps. For instance, we prove that for \(r=3\), the problems \(r\)-Fix-Promise and \(r\)-Swap-Promise are -hard for planar graphs. As a consequence of our reduction, the problems cannot be solved in \(2^{o(\sqrt{n})}\) time unless the Exponential Time Hypothesis fails.