Preprocessing to Reduce the Search Space for Odd Cycle Transversal (2409.00245v2)

Abstract: 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.