Fixing Improper Colorings of Graphs

Abstract

In this paper we consider a variation of a recoloring problem, called the Color-Fixing. Let us have some non-proper r-coloring ϕ of a graph G. We investigate the problem of finding a proper r-coloring of G, which is "the most similar" to ϕ, i.e., the number k of vertices that have to be recolored is minimum possible. We observe that the problem is NP-complete for any fixed r ≥ 3, even for bipartite planar graphs. Moreover, it is W [1]-hard even for bipartite graphs, when parameterized by the number k of allowed recoloring transformations. On the other hand, the problem is fixed-parameter tractable, when parameterized by k and the number r of colors.We provide a 2 n ⋅n O(1) algorithm for the problem and a linear algorithm for graphs with bounded treewidth. We also show several lower complexity bounds, using standard complexity assumptions. Finally, we investigate the fixing number of a graph G. It is the minimum k such that k recoloring transformations are sufficient to transform any coloring of G into a proper one.

On the complexity of restoring corrupted colorings

“…• In Section 4, we prove that under plausible complexity assumptions, r-Fix has no polynomial kernel parameterized by the number of recolorings k, for every r ≥ 3. We stress that while mentioned as an open problem in [20], the question was subsequently answered by Garnero, Junosza-Szaniawski, Liedloff, Montealegre, and Rzążewski in a full version [13] of [20]. Our result was obtained independently of their work, and uses slightly different ideas.…”

“…Junosza-Szaniawski et al [20] focused on the r-Fix problem, that is, the number of colors in the coloring is fixed to be r. Among other results, they showed the problem is FPT parameterized by treewidth.…”

“…Junosza-Szaniawski et al [20] proved that for every fixed r, the problem r-Fix is FPT parameterized by the number of recolorings. However, when the basic operation is a swap instead of a recoloring, Figure 1: An instance (G, k) of Independent Set with k = 2 (highlighted in gray) transformed to an instance of r-Swap.…”

“…Junosza-Szaniawski et al [20] showed that for any fixed r, the problem r-Fix is FPT parameterized by the number of recolorings k. In particular, their result implies a kernel of exponential size for the problem. Thus, they asked whether or not there is a kernel of polynomial size.…”

“…Furthermore, they show that for any fixed r, the problem is fixed-parameter tractable (FPT) parameterized by the number of recolorings k. Finally, in the same paper, the authors show that for graphs of treewidth t, the problem can be solved in O(nr t+2 ) time. For a discussion on several related reoptimization and reconfiguration problems, we refer the reader to [20]. We also note that their results are considerably expanded in a full version of the manuscript currently under review [13].…”