The Parameterized Complexity Landscape of Two-Sets Cut-Uncut

TL;DR

This paper studies the computational complexity of Two-Sets Cut-Uncut, a graph separation problem, analyzing its fixed-parameter tractability and kernelization properties based on various graph parameters.

Contribution

It provides a systematic parameterized complexity analysis of Two-Sets Cut-Uncut, establishing FPT results and kernelization bounds for key graph parameters.

Findings

FPT algorithms for vertex-deletion distance to cographs

No polynomial kernels for vertex cover number under standard assumptions

Near-complete complexity characterization within the parameterized hierarchy

Abstract

In Two-Sets Cut-Uncut, we are given an undirected graph $G=(V,E)$ and two terminal sets $S$ and $T$. The task is to find a minimum cut $C$ in $G$ (if there is any) separating $S$ from $T$ under the following ``uncut'' condition. In the graph $(V,E \setminus C)$, the terminals in each terminal set remain in the same connected component. In spite of the superficial similarity to the classic problem Minimum $s$-$t$-Cut, Two-Sets Cut-Uncut is computationally challenging. In particular, even deciding whether such a cut of any size exists, is already NP-complete. We initiate a systematic study of Two-Sets Cut-Uncut within the context of parameterized complexity. By leveraging known relations between many well-studied graph parameters, we characterize the structural properties of input graphs that allow for polynomial kernels, fixed-parameter tractability (FPT), and slicewise polynomial algorithms (XP). Our main contribution is the near-complete establishment of the complexity of these algorithmic properties within the described hierarchy of graph parameters. On a technical level, our main results are fixed-parameter tractability for the (vertex-deletion) distance to cographs and an OR-cross composition excluding polynomial kernels for the vertex cover number of the input graph (under the standard complexity assumption NP is not contained in coNP/poly).