Preprocessing to Reduce the Search Space for Odd Cycle Transversal
The NP-hard Odd Cycle Transversal problem asks for a minimum vertex set whose removal from an undirected input graph $G$ breaks all odd cycles, and thereby yields a bipartite graph. The problem is well-known to be fixed-parameter tractable when parameterized by the size $k$ of the desired solution....
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Zusammenfassung: | The NP-hard Odd Cycle Transversal problem asks for a minimum vertex set whose
removal from an undirected input graph $G$ breaks all odd cycles, and thereby
yields a bipartite graph. The problem is well-known to be fixed-parameter
tractable when parameterized by the size $k$ of the desired solution. It also
admits a randomized kernelization of polynomial size, using the celebrated
matroid toolkit by Kratsch and Wahlstr\"{o}m. The kernelization guarantees a
reduction in the total $\textit{size}$ of an input graph, but does not
guarantee any decrease in the size of the solution to be sought; the latter
governs the size of the search space for FPT algorithms parameterized by $k$.
We investigate under which conditions an efficient algorithm can detect one or
more vertices that belong to an optimal solution to Odd Cycle Transversal. By
drawing inspiration from the popular $\textit{crown reduction}$ rule for Vertex
Cover, and the notion of $\textit{antler decompositions}$ that was recently
proposed for Feedback Vertex Set, we introduce a graph decomposition called
$\textit{tight odd cycle cut}$ that can be used to certify that a vertex set is
part of an optimal odd cycle transversal. While it is NP-hard to compute such a
graph decomposition, we develop parameterized algorithms to find a set of at
least $k$ vertices that belong to an optimal odd cycle transversal when the
input contains a tight odd cycle cut certifying the membership of $k$ vertices
in an optimal solution. The resulting algorithm formalizes when the search
space for the solution-size parameterization of Odd Cycle Transversal can be
reduced by preprocessing. To obtain our results, we develop a graph reduction
step that can be used to simplify the graph to the point that the odd cycle cut
can be detected via color coding. |
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DOI: | 10.48550/arxiv.2409.00245 |