graph cut-set analysis

This work addresses the critical challenge of efficiently identifying minimum cut sets (MCS) between source–destination pairs in large-scale complex networks. To this end, we propose Fast-MCS, a scalable open-source tool grounded in graph theory and combinatorial optimization. By optimizing algorithms for cut-set enumeration and verification, and integrating efficient data structures with parallelized memory management, Fast-MCS achieves substantial improvements in computational efficiency. Experimental evaluations on multiple large network instances demonstrate that Fast-MCS significantly reduces MCS computation time compared to state-of-the-art methods, offering a practical and highly efficient solution for reliability analysis in complex networks.

This work addresses the high computational complexity of cut-set computation in multi-path ensemble attribute evaluation by proposing an efficient algorithm and developing a vectorized computing framework based on matrix operations, which reformulates path attribute calculations as parallelizable array operations. For the first time, this approach provides a practical implementation of the formal model for path set attributes, integrating an optimized cut-set algorithm with array-oriented programming languages to substantially improve computational efficiency. Empirical evaluations across network simulations of varying complexity demonstrate that the method yields predictable and acceptable execution times, thereby establishing a practical foundation for large-scale multi-path analysis.

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.

This paper investigates the parameterized complexity of the Multicut problem on embedded graphs, focusing on how the structure of the demand graph $H$ affects computational efficiency. Specifically, it addresses finding a minimum-weight edge cut in $G$ that separates all adjacent terminal pairs in $H$. The work establishes the first tight characterization linking demand graph classes to the exponential base of algorithmic time complexity: only when $H$ is close to a complete bipartite graph can the problem be solved in $2^{o(sqrt{n})}$ time, breaking prior lower bounds; otherwise, a matching $2^{Omega(sqrt{n})}$ lower bound holds. The approach integrates graph embedding theory, crossing-number arguments, structured dynamic programming, and closure-property-driven parameterized analysis. The results unify and generalize several important special cases—including Multiway Cut and Group 3-Terminal Cut—and establish their optimality under the Exponential Time Hypothesis.

This paper addresses the graph dismantling problem under edge-budget constraints for spatially embedded planar graphs (e.g., transportation and power grids), aiming to quantify the impact of edge removal on network connectivity robustness. We propose a “spanning-tree skeleton–dual-path” framework: multiple spanning trees are uniformly sampled to construct a structural skeleton; combined with logarithmic density feature estimation and a slope-prediction model, the framework adaptively selects either fine-grained dismantling (for small budgets) or rapid fragmentation (for large budgets). The method ensures both interpretability and computational efficiency, achieving near-linear time complexity on random planar graphs. It significantly reduces the size of the largest connected component and uncovers a clear quantitative relationship between edge budget and fragmentation extent. This work establishes a novel paradigm for robustness assessment of critical infrastructure networks.

This work addresses the long-standing challenge of achieving subquadratic query complexity for fundamental problems such as reachability in directed graphs under the cut-query model. Focusing on directed acyclic graphs (DAGs), the authors combine divide-and-conquer techniques with structural properties of DAGs to design the first algorithm with nearly linear query complexity. Specifically, single-source reachability and topological ordering can be determined using $O(n \log^3 n)$ cut queries. The approach is further extended to detect cycles in general directed graphs within the same asymptotic query bound. This result breaks a longstanding efficiency barrier in the cut-query model for directed graphs, marking a significant theoretical advance in the field.

Existing resources for computational complexity reductions lack systematic organization, visualization, and extensibility. Method: This paper introduces an interactive, graph-database–based online platform that unifies diverse reduction paradigms—including classical complexity classes (e.g., NP, #P), parameterized classes (W[1], W[2]), gap-preserving reductions, and PCP theorems—within a single directed graph model linking problems and reductions. The platform supports semantic search, multi-dimensional filtering, and community-driven curation, and employs a modular software architecture enabling seamless integration of new complexity classes and reduction types. Contribution/Results: The platform publicly releases an interconnected knowledge graph covering core classes such as NP, #P, SSP-NP, W[1], W[2], and PCP. It establishes an open, verifiable, evolvable, and reusable knowledge infrastructure for complexity theory research, advancing both pedagogy and collaborative investigation of reduction hierarchies.

This work addresses the limitation of existing exact solvers for large-scale Maximum k-Cut problems (k > 2), which stems from the absence of effective preprocessing techniques. The paper introduces, for the first time, optimality-preserving data reduction rules tailored to this problem, leveraging structured cutset identification and graph decomposition strategies to partition the input graph into independently solvable connected components. A novel proof framework based on weighted graph superposition is developed to underpin these reductions. By engineering an integration of established MaxCut preprocessing methods into a unified system, the authors present the first efficient preprocessing pipeline specifically designed for k > 2. Experimental results demonstrate that the proposed approach substantially reduces instance sizes, significantly accelerates exact solvers when integrated, and enables solving more instances to optimality than previously possible.

This study addresses the problem of maintaining k-edge connectivity in dynamic undirected graphs under single-edge insertions or deletions. It is the first to model this task as a structural regulation problem, proposing an active maintenance framework that integrates redundant edge elimination and connectivity restoration. The approach combines Nagamochi–Ibaraki sparse certificates, Link-Cut Trees, and Dinic’s algorithm to achieve efficient updates on sparse graphs. A key innovation is the introduction of a local augmentation strategy that avoids trivial fallbacks. Theoretical analysis shows that redundant edge elimination can be performed in O(k log n) amortized time, while connectivity restoration completes in O(k·n^{5/3}) time, significantly improving the efficiency of dynamic k-edge connectivity maintenance.

This study addresses the problem of efficiently enumerating minimal removable vertex sets (MinRS) whose removal triggers core collapse in k-cores, thereby assessing network robustness under node deletions. Building upon a monotone system framework, the authors propose a general method to enumerate all MinRS for any k-core variant. The key innovation lies in the first formal definition of the "in-dominating seed property" and the proof that standard k-cores satisfy this property, which reduces the enumeration complexity from O((n+m)n) to O((n+m)log n). This yields the first O((n+m)log n)-delay algorithm for enumerating all k-core subgraphs. The approach naturally extends to weighted, multilayer, and (k,ℓ)-core models, significantly outperforming the baseline algorithm by Boley et al. (2010).