## On the complexity of restoring corrupted colorings

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**On the complexity of restoring corrupted colorings.** / De Biasi, Marzio; Lauri, Juho.

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*Journal of Combinatorial Optimization*, vol. 37, no. 4, pp. 1150-1169. https://doi.org/10.1007/s10878-018-0342-2

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*Journal of Combinatorial Optimization*,

*37*(4), 1150-1169. https://doi.org/10.1007/s10878-018-0342-2

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TY - JOUR

T1 - On the complexity of restoring corrupted colorings

AU - De Biasi, Marzio

AU - Lauri, Juho

PY - 2019/5

Y1 - 2019/5

N2 - 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 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≥ 3 , the problem r-Fix does not admit a polynomial kernel unless [InlineEquation not available: see fulltext.]. 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 [InlineEquation not available: see fulltext.]-hard whereas r-Fix is known to be FPT. Moreover, when r is part of the input, we observe both Fix and Swap are [InlineEquation not available: see fulltext.]-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 [InlineEquation not available: see fulltext.]-hard for planar graphs. As a consequence of our reduction, the problems cannot be solved in 2o(n) time unless the Exponential Time Hypothesis fails.

AB - 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 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≥ 3 , the problem r-Fix does not admit a polynomial kernel unless [InlineEquation not available: see fulltext.]. 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 [InlineEquation not available: see fulltext.]-hard whereas r-Fix is known to be FPT. Moreover, when r is part of the input, we observe both Fix and Swap are [InlineEquation not available: see fulltext.]-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 [InlineEquation not available: see fulltext.]-hard for planar graphs. As a consequence of our reduction, the problems cannot be solved in 2o(n) time unless the Exponential Time Hypothesis fails.

KW - Combinatorial reconfiguration

KW - Computational complexity

KW - Graph coloring

KW - Local search

KW - Parameterized complexity

U2 - 10.1007/s10878-018-0342-2

DO - 10.1007/s10878-018-0342-2

M3 - Article

VL - 37

SP - 1150

EP - 1169

JO - Journal of Combinatorial Optimization

JF - Journal of Combinatorial Optimization

SN - 1382-6905

IS - 4

ER -