The Parameterized Complexity Landscape of Two-Sets Cut-Uncut

TL;DR

Two-Sets Cut-Uncut asks for a small $S$-$T$ cut that preserves connectivity within $S$ and $T$. The authors provide a structured parameterized-complexity landscape, delivering a cotree-based DP yielding $FPT$ results for distance to cographs and presenting both positive (kernel) and negative (kernel lower bounds) results via OR-cross-compositions, along with XP algorithms for other structural parameters. The work establishes a near-complete tetrachotomy across kernelizability, fixed-parameter tractability, and XP-time algorithms, clarifying when preprocessing and FPT methods are feasible for this problem. It also identifies key open questions (e.g., distance to interval graphs, clique-width FPT status, and the number of terminals) that guide future research on the parameterized complexity of Two-Sets Cut-Uncut.

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).

Figures (6)

• Figure 1: Overview of our results. An edge between two parameters $\alpha$ and $\beta$, where $\alpha$ is above $\beta$, indicates that in any instance, the value of $\beta$ is upper-bounded by a function only depending on the value of $\alpha$. Any hardness result for $\alpha$ immediately implies the same hardness result for $\beta$ and any positive result for $\beta$ immediately implies the same positive result for $\alpha$ (where we additionally require that the dependency is polynomial if we show or exclude a polynomial kernel). Green boxes indicate the existence of polynomial kernels, yellow boxes show that the parameter admits fixed-parameter tractability but no polynomial kernel, an orange box indicates polynomial-time algorithms for constant parameter values (XP) but no fixed-parameter tractability, and a red box shows that the parameter is NP-hard for some constant parameter value. We mention that the status of Two-Sets Cut-Uncut parameterized by distance to interval graphs, number of terminals (XP/para-NP-hard), and clique-width (fixed-parameter tractable/W[1]-hard) remain open.
• Figure 2: Illustration of the reduction by Bentert et al.. The dashed edges represent $n+2m$ parallel $P_3$'s as indicated between $s$ and $v_1$.
• Figure 3: An illustration of the reduction behind \ref{['prop:md']} with $C_1 = (x_1 \lor x_2 \lor \overline{x}_n)$.
• Figure 4: An illustration of the construction in the proof of \ref{['thm:nopolyvc']}. The solid boxes indicate cliques and the dashed boxes indicate independent sets.
• Figure 5: An illustration of the reduction behind \ref{['prop:nopolysol']}.

Theorems & Definitions (18)

• Definition 1: OR-cross-composition BJK14
• Theorem 1
• proof
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Theorem 2