fpt motif counting

This study investigates the fine-grained complexity of exactly counting three-hyperedge motifs in hypergraphs that realize arbitrary Venn diagram intersection patterns, parameterized by hypergraph rank. Leveraging the Triangle Detection and Hyperclique hypotheses within the framework of parameterized complexity theory and combinatorial analysis, the work provides the first complete characterization of the complexity landscape for all such patterns. It establishes that a fixed-parameter near-linear time algorithm exists if and only if the Venn diagram corresponds to a degenerate case—specifically, when one hyperedge is entirely contained within another. For all non-degenerate patterns, the problem provably requires nearly quadratic time. This result yields a comprehensive and precise map of the computational complexity of motif counting under rank parameterization.

This paper addresses the efficient counting of small pattern subhypergraphs in bounded-degeneracy hypergraphs. To overcome the inefficiency of traditional algorithms on complex hypergraphs, we propose a unified hypergraph degeneracy framework that— for the first time—precisely characterizes the class of subhypergraph patterns admitting near-linear-time solutions: namely, those excluding certain “forbidden patterns.” Leveraging this framework and integrating combinatorial enumeration with structural decomposition techniques, we design an exact counting algorithm with time complexity $O(n log n)$ under fine-grained complexity assumptions. Moreover, for patterns containing forbidden structures, we establish tight computational lower bounds, thereby yielding the first complete complexity classification for subhypergraph counting. Our results provide both theoretical foundations and practical algorithmic tools for higher-order relational analysis in network science and database systems.

This work addresses the subgraph counting problem on bounded-degeneracy graphs, focusing on arbitrary pattern graphs with at most nine vertices and cycles of length at most ten. We propose the first unified framework that reduces subgraph counting on degenerate graphs to counting small hypergraphs, integrating degeneracy ordering, hierarchical sampling, and hypergraph reduction techniques. Our method achieves the first subquadratic-time algorithms for counting any pattern graph on ≤9 vertices, with time complexity $ ilde{O}(n^{5/3})$, and extends subquadratic-time cycle counting to all cycles of length ≤10—resolving a long-standing open case. This establishes the first systematic subquadratic algorithmic theory for subgraph counting on bounded-degeneracy graphs, significantly extending the applicability boundary of the classical Chiba–Nishizeki paradigm.

Approximating the count and uniformly sampling Hamiltonian motifs (i.e., connected subgraphs containing all vertices of a given subset) in large graphs within sublinear time—under the standard query model supporting only degree, adjacency, and pair queries—has remained open, with prior work restricted to radius-1 motifs (e.g., edges, stars, cliques). Method: We propose a unified framework based on uniform sampling, integrating hierarchical path enumeration with importance-weighted estimation to bridge the “scope gap” between standard and augmented query models. Contribution/Results: This is the first algorithm achieving ε-approximate counting and approximately uniform sampling for *arbitrary* Hamiltonian motifs under the standard query model. It runs in Õ(n + m) time—sublinear in graph size—and simultaneously simplifies and unifies the design and analysis of algorithms for classical motifs including stars, triangles, and k-cliques, significantly reducing both theoretical complexity and implementation overhead.

This work addresses the efficient (1+ε)-approximation of the number of occurrences of permutation patterns of length k ≤ 5 in real-valued sequences. While exact counting is computationally prohibitive, we present the first deterministic near-linear-time algorithm with time complexity O(n log n / ε²). Methodologically, we introduce Birgé’s distribution decomposition—previously unexplored in permutation pattern counting—integrated with a divide-and-conquer framework and discrete geometric embedding. This synergy enables the first provable separation between approximate and exact counting complexities. Our approach breaks known lower-bound barriers for k ≤ 5 and, empirically, achieves significantly faster runtime than exact algorithms for k = 4. Beyond improving asymptotic efficiency, this work pioneers the application of distribution testing techniques to combinatorial pattern counting, opening a new methodological avenue at the intersection of property testing, computational geometry, and enumerative combinatorics.

This work investigates the fine-grained complexity of approximately counting occurrences of a length-$k$ permutation pattern within a length-$n$ permutation. Under the Exponential Time Hypothesis (ETH), we establish via fine-grained reductions that no algorithm running in time $f(k) \cdot n^{o(k/\log k)}$ can approximate the count within a multiplicative error of $n^{(1/2 - \varepsilon)k}$ for any $\varepsilon > 0$. This result refutes the conjecture that approximate counting of small patterns is significantly easier than exact counting, and it provides the first conditional equivalence between approximate and exact counting in this setting. Moreover, our lower bound yields an almost tight trade-off between approximation error and running time, nearly matching the known upper bound of $n^{k/2}$.

This work addresses #P-hard counting problems—such as counting independent sets in general graphs and #2-SAT—that are inapproximable in polynomial time and prohibitively expensive to solve exactly. The authors propose a novel framework based on bounded, unweighted self-reducibility, which recursively decomposes problem instances and aggregates upper bounds from subproblems at a square-root recursion depth. By integrating enumeration with a hybrid sampling estimator, the approach substantially reduces the base of the exponential time complexity. The method achieves improved runtimes of O*(1.1869ⁿ) for independent set counting and O*(1.2373ⁿ) for #2-SAT approximation, outperforming the best known exact algorithms. It further extends to counting maximum cliques, minimal separators, and perfect matchings in subcubic graphs, and admits black-box quantum speedup.

This work investigates the homomorphism counting problem for temporal patterns with partially ordered edges in large temporal graphs, aiming to characterize the structural expressiveness of temporal graphs. By establishing an equivalence between temporal graph isomorphism and homomorphism counts of temporal patterns, the study formulates a temporal analogue of Lovász’s isomorphism theorem. It introduces a novel width parameter, termed *toadwidth*, to analyze fixed-parameter tractability. Combining techniques from parameterized complexity, extensions of clique-width, and combinatorial graph theory, the paper proves that two temporal graphs are isomorphic if and only if they admit identical homomorphism counts for all temporal patterns; that homomorphism counting is fixed-parameter tractable when parameterized by bounded toadwidth; and provides a sharp dichotomy criterion for the parameterized complexity of homomorphism counting in the case of totally ordered temporal patterns.

本文提出了一种无参数的流式三角计数算法，解决了在不知道三角形数量T的情况下，如何在亚线性空间内近似计算图中的三角形数量问题。

This study addresses the computational intractability of large-scale motif discovery in networks with high-degree hub nodes, where combinatorial explosion severely limits existing approaches. To overcome this, we propose a fast scanning strategy that integrates k-core decomposition with isomorphic subtree counting, achieving efficient subgraph enumeration by prioritizing the traversal of peripheral network structures. Furthermore, we establish from a complexity-theoretic perspective that the key subroutine is #P-complete. This work breaks through the motif size limitations inherent in current algorithms. Experiments conducted on eleven real-world networks demonstrate that the proposed method substantially reduces computational overhead and improves solving efficiency, proving particularly effective for complex real-world networks characterized by rich peripheral structures.