FPT Algorithms for Conflict-free Coloring of Graphs and Chromatic Terrain Guarding

We present fixed parameter tractable algorithms for the conflict-free coloring problem on graphs. Given a graph $G=(V,E)$, \emph{conflict-free coloring} of $G$ refers to coloring a subset of $V$ such that for every vertex $v$, there is a color that is assigned to exactly one vertex in the closed neighborhood of $v$. The \emph{k-Conflict-free Coloring} problem is to decide whether $G$ can be conflict-free colored using at most $k$ colors. This problem is NP-hard even for $k=1$ and therefore under standard complexity theoretic assumptions, FPT algorithms do not exist when parameterised by the solution size. We consider the \emph{k-Conflict-free Coloring} problem parameterised by the treewidth of the graph and show that this problem is fixed parameter tractable. We also initiate the study of \emph{Strong Conflict-free Coloring} of graphs. Given a graph $G=(V,E)$, \emph{strong conflict-free coloring} of $G$ refers to coloring a subset of $V$ such that every vertex $v$ has at least one colored vertex in its closed neighborhood and moreover all the colored vertices in $v$'s neighborhood have distinct colors. We show that this problem is in FPT when parameterised by both the treewidth and the solution size. We further apply these algorithms to get efficient algorithms for a geometric problem namely the Terrain Guarding problem, when parameterised by a structural parameter.