graph algorithms

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 systematically investigates the computational complexity of three fundamental problems in Hierarchical Task Network (HTN) planning: plan verification, executability checking, and state reachability. Using structured graph-theoretic modeling, parameterized algorithm design, and tight lower-bound constructions, it establishes the first complete W[1]/FPT classification of these problems under standard parameters. It introduces the first meta-theorem that lifts polynomial-time solvability from primitive task networks to general task networks, and proves the tightness of its preconditions. Furthermore, for natural classes of primitive networks, it derives new polynomial-time algorithms for all three problems and provides matching conditional lower bounds. These results constitute a systematic breakthrough in HTN complexity theory, enabling principled complexity transfer across HTN formalisms and unifying previously fragmented analyses.

This work addresses real-time network connectivity planning scenarios where tree construction and execution must interleave dynamically, overcoming the high preprocessing overhead and response latency inherent in conventional two-phase MST algorithms. We propose an *edge-enumeration* approach to MST construction, introducing the first unified theoretical framework for efficient edge enumeration across three graph classes: undirected unweighted, directed, and weighted graphs. We derive tight delay bounds for enumeration and prove that meaningful edge enumeration is impossible for weighted directed graphs. Our method employs traversal-based, priority-queue-driven algorithms explicitly optimized for enumeration delay, incorporating adversarial lower-bound analysis and degree-aware delay modeling. On undirected unweighted graphs, it achieves average-case output delay proportional to the graph’s average degree. Experiments demonstrate substantially lower preprocessing cost and significantly reduced first-edge response latency compared to classical MST algorithms, enabling low-latency, real-time connectivity planning in dynamic networks.

This work addresses four #P-complete graph counting problems in computational chemistry—Kekulé structure enumeration, Hosoya index, Merrifield–Simmons index, and matching/independent set entropy—by introducing the first fixed-parameter tractable (FPT) framework parameterized by treewidth (tw) and pathwidth (pw). We design a unified dynamic programming algorithm for these chemical topological indices, achieving time complexity $O^*(2^{O( ext{tw})})$, substantially improving upon existing exponential-time baselines. Theoretical analysis and empirical evaluation on the full PubChem dataset (>100 million compounds) demonstrate that >99.9% of real chemical graphs satisfy $ ext{tw} leq 6$, validating the small-treewidth hypothesis. Our implementation scales efficiently to large-scale chemical graphs, delivering speedups of several orders of magnitude over naive enumeration. This work bridges parameterized algorithms and cheminformatics, establishing a new paradigm for exact computation of #P-hard chemical graph metrics.

This paper addresses the fragmentation and lack of interoperability among structural complexity measures across graph theory, geometric group theory, and dynamical systems. Methodologically, it introduces a unified “structured decomposition” framework grounded in category theory: (i) it is the first to formalize diverse domain-specific decomposition paradigms categorically; (ii) it establishes a general duality theory linking decompositions to object completions; and (iii) it defines composable width functors enabling cross-model quantification, comparison, and translation of structural complexity. Key contributions include: a unified categorical characterization of over ten complexity parameters—including treewidth, layered treewidth, and hypergraph treewidth—revealing their intrinsic structural relationships; and a novel parameterized tractability paradigm for NP-hard problems, grounded in decomposition width. The framework achieves both theoretical unification and algorithmic realizability.

We consider the problem of partitioning the edges of a graph into as few paths as possible. This is a~subject of the classic conjecture of Gallai and a recurring topic in combinatorics. Regarding the complexity of partitioning a graph optimally, Peroch\'e [Discret. Appl. Math., 1984] proved that it is NP-hard already on graphs of maximum degree four, even when we only ask if two paths suffice. We show that the problem is solvable in polynomial time on subcubic graphs and then we present an efficient algorithm for ``almost-subcubic''graphs. Precisely, we prove that the problem is fixed-parameter tractable when parameterized by the edge-deletion distance to a subcubic graph. To this end, we reduce the task to model checking in first-order logic extended by disjoint-paths predicates ($\mathsf{FO}\text{+}\mathsf{DP}$) and then we employ the recent tractability result by Schirrmacher, Siebertz, Stamoulis, Thilikos, and Vigny [LICS 2024].

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.

This work investigates how to accelerate parallel connectivity and shortest-path algorithms by efficiently augmenting a directed acyclic graph with transitive-closure edges to substantially reduce its diameter. The authors introduce a structural criterion called “certified shortcut edges,” which precisely characterizes the set of shortcut edges constructible by any near-linear-time algorithm. Leveraging this criterion in conjunction with combinatorial graph theory and complexity lower-bound techniques, they improve the known diameter lower bound from $n^{2/9 - o(1)}$ to $n^{1/4 - o(1)}$, establishing that no near-linear-time algorithm can achieve a smaller diameter. This result significantly strengthens existing theoretical limits and provides a robust foundation for the design of future efficient shortcut-edge construction methods.

This work proposes an efficient method for atomic decomposition of undirected graphs that circumvents the complex pipeline traditionally reliant on minimal triangulation and minimal clique separators. For the first time, it establishes a theoretical connection between Maximum Cardinality Search (MCS) orderings and atomic decomposition, enabling the direct recursive generation of atoms through the convex extension of node 1 and its neighborhood. By integrating two existing convex extension strategies, the proposed algorithm achieves significantly improved computational efficiency while preserving correctness. Experimental results demonstrate that the method outperforms current state-of-the-art graph decomposition algorithms in terms of runtime performance.

This work addresses the problem of computing the girth—the length of the shortest cycle—in intersection graphs of planar line segments. We present the first algorithm with expected running time $O(n^{1.483})$, breaking the long-standing $O(n^{3/2})$ barrier. Our approach combines subcubic bounded-difference Min-Plus matrix multiplication with a novel variant of planar graph separator theorems. The technique extends to intersection graphs of connected algebraic curves or semi-algebraic sets of constant description complexity. This result resolves an open question posed by Chan at SODA 2023 and significantly advances the theoretical limits for girth computation in geometric intersection graphs.