Connectivity-Preserving Important Separators: Enumeration and an Improved FPT Algorithm for Node Multiway Cut-Uncut

TL;DR

This work extends the classical important-separator paradigm by introducing connectivity-preserving important separators (CPIS), which enforce connectivity constraints on terminal sets while achieving separation. The authors prove a near-single-exponential bound on the number of CPIS of size at most k and provide an enumeration framework running in time proportional to 2^{O(k log k)} · n · T(n,m). As a key application, they derive a substantially faster fixed-parameter algorithm for the Node-Multiway-Cut-Uncut problem, achieving 2^{O(k log k)} · T(n,m) time (and near m^{1+o(1)} overhead for polynomially bounded integer weights), improving the prior 2^{O(k^2 log k)} barrier. The CP framework thus generalizes important-separator techniques to problems with both cut and uncut constraints, offering refined tools for designing FPT algorithms in graph separation tasks.

Abstract

We develop a framework for handling graph separation problems with connectivity constraints. Extending the classical concept of important separators, we introduce and analyze connectivity-preserving important separators, which are important separators that not only disconnect designated terminal sets $A$ and $B$ but also satisfy an arbitrary set of connectivity constraints over the terminals. These constraints can express requirements such as preserving the internal connectivity of each terminal set, enforcing pairwise connections defined by an equivalence relation, or maintaining reachability from a specified subset of vertices. We prove that for any graph $G=(V,E)$, terminal sets $A,B\subseteq V$, and integer $k$, the number of important $A,B$-separators of size at most $k$ satisfying a set of connectivity constraints is bounded by $2^{O(k\log k)}$, and that all such separators can be enumerated within $O(2^{O(k\log k)} \cdot n \cdot T(n,m))$ time, where $T(n,m)$ is the time required to compute a minimum $s,t$-separator. As an application, we obtain a new fixed-parameter-tractable algorithm for the Node Multiway Cut-Uncut (N-MWCU) problem, parameterized by $k$, the size of the separator set. The algorithm runs in $O(2^{O(k\log k)} \cdot n \cdot m^{1+o(1)})$ time for graphs with polynomially-bounded integer weights. This significantly improves the dependence on $k$ from the previous $2^{O(k^2\log k)}$ to $2^{O(k\log k)}$, thereby breaking a long-standing barrier, and simultaneously improves the polynomial factors. Our framework generalises the important-separator paradigm to separation problems in which the deletion set must satisfy both cut and uncut constraints on terminal subsets, thus offering a refined combinatorial foundation for designing fixed-parameter algorithms for cut-uncut problems in graphs.

Figures (1)

• Figure 1: Illustration for the proof of Lemma \ref{['lem:technicalfortworesults']}, case 2. The vertices of $T$ are represented by the black solid vertices. The graph $G'$ is enclosed in the dotted ellipse, and the vertices of $T"$ (that are outside of $G'$) are encircled. The connected components $C_1,C_2,C_3\in \mathcal{C}(G\hbox{[}1.0]{$-$} T)$ have the property that $C_i\cap D\neq \emptyset$.

Theorems & Definitions (56)

• Theorem 1.1
• Theorem 1.2
• Definition 1.1
• Definition 1.2
• Lemma 1.1
• Definition 1.3
• Lemma 1.2
• Lemma 1.3
• Proposition 1.1
• Definition 1.4