Preprocessing to Reduce the Search Space: Antler Structures for Feedback Vertex Set

TL;DR

This paper introduces antler decompositions as a new preprocessing tool for speeding up exact algorithms for NP-hard problems, focusing on undirected Feedback Vertex Set (FVS). By identifying 1-antlers and their generalizations, it provides reductions that certify that certain vertices must lie in any optimal FVS, thereby shrinking the search space for FVS computations. The authors establish both hardness results (NP-hard and W[1]-hard cases) and constructive FPT algorithms using color coding and universal sets to locate antlers or reduce FVCs while preserving optimality. They also present a comprehensive reduction framework that yields a family of rules (FVS-safe and antler-safe) to compress the graph and enable efficient search for large, structured certificates of optimality. Overall, the work lays the groundwork for a new paradigm in preprocessing where solution structure, not just instance size, drives kernelization-like improvements for NP-hard problems.

Abstract

The goal of this paper is to open up a new research direction aimed at understanding the power of preprocessing in speeding up algorithms that solve NP-hard problems exactly. We explore this direction for the classic Feedback Vertex Set problem on undirected graphs, leading to a new type of graph structure called antler decomposition, which identifies vertices that belong to an optimal solution. It is an analogue of the celebrated crown decomposition which has been used for Vertex Cover. We develop the graph structure theory around such decompositions and develop fixed-parameter tractable algorithms to find them, parameterized by the number of vertices for which they witness presence in an optimal solution. This reduces the search space of fixed-parameter tractable algorithms parameterized by the solution size that solve Feedback Vertex Set.

Figures (4)

• Figure 1: Graph structures showing there is an optimal solution containing all blue vertices and no gray vertices, certified by the blue subgraph. Left: Crown decomposition for Vertex Cover. Right: Antler for Feedback Vertex Set. For legibility, the number of edges in the drawing has been restricted. It therefore has vertices of degree at most $2$, which makes the graph it reducible by standard reduction rules; but adding all possible edges between gray and blue vertices leads to a structure of minimum degree at least three which is still a 1-antler.
• Figure 2: An example of a $3$-antler. The subgraph $G'$, marked in blue, has a feedback vertex number of $5$ showing that any feedback vertex set of $G$ contains at least $|\mathsf{head}\xspace| = 5$ vertices from $\mathsf{head}\xspace \cup \mathsf{antler}\xspace$. Each connected component of $G'$ has a feedback vertex set of size at most $3$. The subgraph $G[\mathsf{antler}\xspace]$ is acyclic and each of its connected components has at most one edge to $V(G) \setminus (\mathsf{head}\xspace \cup \mathsf{antler}\xspace)$.
• Figure 3: Consider the $3$-antler $(\mathsf{head}\xspace, \mathsf{antler}\xspace) = (C,F)$ from \ref{['fig:antler:3-antler']}. The pair $(C \setminus X, F)$ remains a $3$-antler after removing a subset $X \subseteq C$ from $G$.
• Figure 5: Standard reduction rules for Feedback Vertex Set reduce any 1-antler of width $1$ to a pair of vertices with two edges between them, one of which has degree $3$. Hence we can reduce the graph until all 1-antlers of width $1$ are removed with the addition of the following reduction rule: If vertices $u$ and $v$ are connected by a double edge and $\mathop{\mathrm{deg}}\nolimits(v) = 3$ then remove $u$ and $v$ from the graph and decrease the solution size by one. These reduction rules can be exhaustively applied in linear time.

Theorems & Definitions (34)

• Theorem 1
• Theorem 2
• Lemma 5
• Lemma 6
• Definition 7
• Lemma 9: DonkersJ21
• Theorem 10
• Corollary 12
• Theorem 13
• Proposition 17