The paper introduces a new graph structure called an antler decomposition, which can be used to identify vertices that belong to an optimal feedback vertex set in a graph.
The key insights are:
An antler decomposition consists of two disjoint vertex sets (head, antler) such that:
The authors show that finding a non-empty antler decomposition is NP-hard, but provide an FPT algorithm to find an antler decomposition parameterized by the size of the head.
They generalize the concept to z-antlers, where the subgraph induced by head and antler contains a head-certificate of order z, meaning each component has a feedback vertex set of size at most z.
The authors provide an FPT algorithm to find a z-antler decomposition parameterized by the size of the head and the order z.
These antler decompositions can be used as a preprocessing step to reduce the search space for exact algorithms solving the Feedback Vertex Set problem, by identifying vertices that must be included in an optimal solution.
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by Huib Donkers... a las arxiv.org 04-12-2024
https://arxiv.org/pdf/2106.11675.pdfConsultas más profundas