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

2025-11-19

AI Summary

This paper studies the graph separation problem under connectivity constraints—specifically, separating a given terminal set while preserving internal connectivity of terminals, inter-class connectivity among equivalence classes, or reachability from a source set. To address this, we introduce the novel concept of *connectivity-preserving important separators*, generalizing classical important separators to arbitrary terminal connectivity constraints and overcoming a long-standing parameter-dependence bottleneck. Building on this framework, we develop a unified algorithm that integrates minimal cut enumeration with fixed-parameter tractable (FPT) tracing techniques. Our algorithm enumerates all valid constrained separators in $2^{O(k log k)}$ time and improves the running time for the Node-weighted Multiway Cut with Uniqueness (N-MWCU) problem from $2^{O(k^2 log k)}$ to $O(2^{O(k log k)} cdot n cdot m^{1+o(1)})$, where $k$ is the solution size, $n$ the number of vertices, and $m$ the number of edges.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Bridging structured and unstructured data

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 $A,Bsubseteq V$, and integer $k$, the number of important $A,B$-separators of size at most $k$ satisfying a set of connectivity constraints is bounded by $2^{O(klog k)}$, and that all such separators can be enumerated within $O(2^{O(klog k)} cdot n cdot T(n,m))$ 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 $O(2^{O(klog k)} cdot n cdot m^{1+o(1)})$ time for graphs with polynomially-bounded integer weights. This significantly improves the dependence on $k$ from the previous $2^{O(k^2log k)}$ to $2^{O(klog k)}$, 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.

Problem

• Develop connectivity-preserving important separators for graph separation problems
• Enumerate all constrained separators with improved FPT algorithm complexity
• Solve Node Multiway Cut-Uncut with enhanced fixed-parameter tractable approach

Innovation

• Introduces connectivity-preserving important separators with constraints
• Enumerates separators in 2^O(k log k) time via new framework
• Improves FPT algorithm for Node Multiway Cut-Uncut problem