Preprocessing to Reduce the Search Space for Odd Cycle Transversal

TL;DR

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 is developed and formalizes when the search space for the solution-size parameterization of Odd Cycle Transversal can be reduced by preprocessing.

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 Wahlström. 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.

Figures (4)

• Figure 1: Examples of crown decomposition (left), antler decomposition for Feedback Vertex Set (middle) and a tight OCC for Odd Cycle Transversal (right). Packings of forbidden subgraphs are highlighted in bold.
• Figure 2: An illustration of the auxiliary graph used in the proof of \ref{['lem:marked:separators:for:all:ARN']}. The left figure shows an example graph $G$ with the given OCC $(X_B,X_C,X_R)$, disjoint sets $C_1, C_2 \subseteq X_C$, a proper $2$-coloring $f_B:X_B \to \{0,1\}$, and another (not necessarily proper) $2$-coloring $f_C:C_1 \to \{0,1\}$. Possibly overlapping terminals $A,R,N$ are determined as described in the proof. The right shows the auxiliary graph $G'$ for $G$, constructed from a copy of $G[X_B]$ with additional $2|X_C|$ vertices. Terminals are partitioned into $(A',R',N',T_X)$, where $T_X$ is deleted when we examine restricted $3$-way cuts. For both figures, the minimum $3$-way separators (in $G[X_B]$ and $G'-T_X$ (restricted), resp.) are shaped in double circles.
• Figure 3: Gadgets used for the proof of \ref{['lem:lb-oct-reduction']}. The left shows a variable gadget, including two vertices representing literals. The right shows a clause gadget, where $s_1$, $s_2$, $s_3$ are connected to corresponding literals.
• Figure 4: A visualization of an auxiliary graph constructed from an instance $(G,k)$ of Multicolored Clique. The set $U$ consists of $n(k-1)$ vertices representing the vertices in $G$ and their adjacent colors. The set $W$ consists of $4m$ vertices representing the edges in $G$ replaced by a gadget for creating odd cycles.

Theorems & Definitions (44)

• Theorem 1
• Lemma 1
• Lemma 2: JansenK21
• Theorem 3: KratschW20
• Lemma 3
• Definition 4: Odd Cycle Cut
• Definition 6: ($z$-)tight OCC
• Lemma 6
• Definition 7
• Lemma 7