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) → [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, Liedloff, and Rzążewski [SOF-SEM 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 ≥ 3, the problem r-Fix does not admit a polynomial kernel unless NP ⊆ coNP/poly. 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 ≥ 3, the problem r-Swap is W[1]-hard whereas r-Fix is known to be FPT. Moreover, when r is part of the input, we observe both Fix and Swap are W[1]-hard parameterized by treewidth. 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 NP-hard for planar graphs. As a consequence of our reduction, the problems cannot be solved in 2 o( √ n) time unless the Exponential Time Hypothesis (ETH) fails.

On Solution Discovery via Reconfiguration

“…The Coloring Discovery problem has already been studied in the color flipping model and the color swapping model under the names k-Fix [23] and k-Swap [3], respectively. Garnero et al [23] show that, for color flipping, the discovery variant is NP-complete, even for bipartite planar graphs.…”

“…Garnero et al [23] show that, for color flipping, the discovery variant is NP-complete, even for bipartite planar graphs. Moreover, they show that the problem is W[1]-hard when parameterized by b, even for bipartite graphs, whereas it is fixed-parameter tractable when parameterized by k + b. Interestingly, the latter is not true in the color swapping model, where the problem is W[1]-hard for any fixed k ≥ 3 when parameterized by b [3]. It is also shown that both problems k-Fix and k-Swap are W[1]hard when parameterized by the treewidth of the input graph.…”