spanning tree algorithms

This work addresses the challenge of efficiently maintaining a directed minimum spanning tree (DMST) under dynamic updates such as edge weight modifications, insertions, or deletions. The authors propose a dynamic algorithm grounded in weighted matroid intersection, which constructs and incrementally maintains an auxiliary graph to iteratively refine the current solution toward the updated DMST, guaranteeing monotonic improvement in solution quality. The key innovation lies in an efficient mechanism for updating the auxiliary graph coupled with a carefully designed iterative optimization strategy that balances theoretical convergence guarantees with practical computational efficiency. Experimental results demonstrate that the proposed method significantly outperforms existing baselines across various dynamic scenarios, offering both superior effectiveness and runtime performance.

This work addresses the challenge of efficiently constructing spanning trees via local computation algorithms (LCAs) in general graphs, where sublinear-time methods—particularly for expander graphs—have been lacking. The authors present the first sublinear-time LCA for high-conductance expanders that locally determines, for any given edge, whether it belongs to a (minimum) spanning tree without globally constructing the entire tree. Their approach combines local probing, conductance-based graph analysis, and random edge weighting, thereby overcoming prior limitations restricted to sparse subgraphs and extending to the weighted minimum spanning tree setting. For an expander with conductance at least φ and maximum degree at most d, the query complexity is O(√n (log²n/φ² + d)); on G(n,p) random graphs, it achieves Õ(√n^{1−δ}), and for minimum spanning trees, the complexity is Õ(√n d²).

This work addresses structural distortion in spanning tree generation for unweighted networks. Traditional Prim’s and Kruskal’s algorithms lack theoretical justification on unweighted graphs, while DFS often yields highly imbalanced, deep trees. We systematically evaluate the applicability of classical algorithms and propose a novel BFS-based spanning tree construction framework that inherently preserves shortest-path distances between nodes and network diameter, while yielding an approximately power-law degree distribution—balancing topological fidelity and tree balance. Extensive experiments across over one thousand real-world and synthetic networks demonstrate that BFS-generated spanning trees significantly outperform baseline methods in both distance preservation and compactness. Our approach provides a theoretically grounded, computationally efficient tool for network backbone extraction, simplified sampling, and structural analysis.

This study investigates the Minimum Covering Spanning Tree (MCST) problem: given a graph $G$ and an integer $k$, determine whether $G$ admits a spanning tree with vertex cover number at most $k$. Methodologically, we establish computational equivalence between MCST and the Dominating Set problem on graphs of diameter at most two and on $P_5$-free graphs—resolving a long-standing open question. We further prove NP-completeness on bipartite planar graphs and unit disk graphs, thereby delineating the precise complexity boundary. Algorithmically, we design a fixed-parameter tractable (FPT) algorithm parameterized by clique-width, and present the first linear-time exact algorithm for interval graphs. Collectively, our work unifies theoretical connections between covering-type spanning trees and classical domination structures in structural graph algorithms, and advances the paradigm of synergistic exploitation of parameterized techniques and graph class properties.

The graph realization problem asks whether a given {0,1}-matrix is the path-incidence matrix of some generating forest—a characterization equivalent to recognizing network matrices, a fundamental subclass of totally unimodular (TU) matrices with critical applications in mixed-integer programming. Existing algorithms adopt column-wise incremental construction, suffering from limited submatrix recognition scope and ambiguity arising from multiple realizations. This paper introduces the first efficient row-wise incremental algorithm: it uniquely characterizes graphic matrices via SPQR trees to resolve realization ambiguity; designs data structures compatible with the Bixby–Wagner column-wise framework; and enables synergistic row-wise extension and column-wise computation for arbitrary submatrix detection. The method significantly improves decision efficiency—especially for matrices admitting multiple realizations—and provides a novel computational tool for leveraging TU structure in optimization.

This work addresses the challenge of efficiently maintaining breadth-first search (BFS) structures under dynamic edge insertions and deletions in directed graphs. It presents the first fully dynamic BFS framework that simultaneously preserves the BFS spanning tree, node levels, and BFS ordering. By integrating novel dynamic data structures with incremental update strategies, the approach cohesively handles the impact of edge modifications on single-source shortest paths and reachability. This study bridges a critical gap in the theoretical understanding of fully dynamic BFS for directed graphs and provides both foundational insights and practical tools for dynamic graph algorithms.

This study addresses the problem of efficiently enumerating all tuples of $k$ edge-disjoint spanning trees in a graph. Building upon Kaiser’s constructive proof of the Tutte–Nash-Williams theorem, the authors propose a recursive algorithm that integrates decision tree construction, depth-first search, and a variant of Kaiser’s technique, dynamically maintaining a forest-packing structure to decompose subproblems. This work presents the first polynomial-delay enumeration algorithm for tuples of edge-disjoint spanning trees, thereby transforming a classical structural result in graph theory into a practical and efficient enumeration method. The algorithm guarantees that all feasible solutions are output completely, with only polynomial delay between successive outputs.

This work addresses the limited accessibility and reproducibility of Bock’s original 1971 algorithm for non-projective dependency parsing, which has hindered its adoption due to its obscure exposition and unclear structure. By performing a line-by-line analysis of the original algorithm, the study provides the first complete execution trace on Bock’s canonical ten-node example and introduces a structured reformulation that explicitly delineates phase structure, state maintenance, and control flow. Furthermore, it unifies the presentation by transforming the maximum-weight arborescence problem into an equivalent minimum-cost formulation via an affine transformation. The resulting reconstruction not only enables full reproducibility and visualization of Bock’s algorithm but also confirms its correctness as an exact decoder for non-projective dependency graphs, substantially enhancing its readability, pedagogical utility, and potential for modern applications.

本文研究了在局部度约束下计算最小生成树的问题，通过不同参数化方法（如树深）探讨了三种形式问题的复杂性差异。

This study addresses the decision, counting, and enumeration problems for interconnected structures in multipartite graphs that satisfy matching constraints and whose projection forms a spanning tree—referred to as “interconnection trees.” The work formally introduces the notion of interconnection trees and proves that their decision problem is NP-complete. However, when the number of partitions is fixed, it presents a fixed-parameter tractable (FPT) algorithm and achieves polynomial- or even linear-time solutions on complete and quasi-complete multipartite graphs. Leveraging parameterized complexity analysis, graph-theoretic algorithm design, and the flashlight search framework, the paper proposes an enumeration algorithm with optimal delay and enhances its practicality through a weight-guided heuristic strategy.