On the complexity of restoring corrupted colorings

Abstract

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.