extremal graph construction

This paper investigates the computational complexity of determining the size $ mathrm{ex}_F(G) $ of a maximum $ F $-free subhypergraph of a given $ k $-uniform hypergraph $ G $, focusing on characterizing when this problem is NP-hard for fixed $ k $-uniform $ F $. Method: To circumvent the absence of a hypergraph Turán theorem, the authors develop a novel hardness characterization framework for edge-modification problems in hypergraphs, integrating combinatorial reductions, algorithmic applications of the Erdős–Ko–Rado theorem, and structural analysis of matchings. Contribution/Results: They establish the first dichotomy: computing $ mathrm{ex}_F(G) $ is NP-hard if $ F $ is non-$ k $-partite, but polynomial-time solvable if $ F $ is a fixed-size matching. This yields a precise conjectural classification of the problem’s complexity and provides the first systematic characterization of computational hardness in extremal hypergraph theory—resolving a long-standing open question.

This paper investigates the computational complexity of detecting minimal hypercycles and answering subgraph queries in hypergraphs and databases, unifying worst-case and average-case hardness analyses. Methodologically, it introduces the first worst-case-to-average-case reduction framework for hypergraph substructure detection; establishes tight (matching) upper and lower bounds for minimal hypercycle detection; designs improved algorithms for detecting long hypercycles; and proves—novelty—the existence of a strict average-case lower bound for hypercycle counting and its equivalent database query problem on random hypergraphs, thereby establishing their average-case hardness. The study integrates combinatorial complexity analysis, hypergraph algorithm design, and probabilistic modeling. Its contributions provide foundational complexity characterizations for hypergraph theory and database query optimization, bridging structural hypergraph properties with practical query evaluation complexity.

This work addresses the challenge in extremal graph theory that the explosive growth in the number of graphs with increasing order hinders systematic exploration of inequalities and extremal structures. The authors propose an exact geometric approach: embedding all non-isomorphic graphs of order up to ten into a two-dimensional invariant space and automatically uncovering extremal graphs and valid inequalities via convex hull computation. They construct a complete database of such non-isomorphic graphs and provide an interactive web platform with API support, enabling conjecture verification and pedagogical applications. Requiring no heuristic assumptions, this method successfully reproduces and extends several classical results and has been integrated into university curricula, significantly enhancing both research efficiency and teaching effectiveness in extremal graph theory.

This study addresses the problem of finding a strong maximum independent set in hypergraphs—defined as a vertex set containing at most one vertex from each hyperedge—which arises in applications such as the construction of perfect minimal hash functions. The work introduces, for the first time, nine specialized data reduction rules grounded in structural properties of hypergraphs, serving as a preprocessing phase to substantially shrink instance sizes. Empirical evaluation demonstrates that this approach reduces instances to an average of 22% of their original size within just 6.76 seconds of preprocessing time. When integrated with the state-of-the-art exact solver, the method yields an average speedup of 3.84×, with peak acceleration reaching 53×, and successfully solves a previously intractable instance.

This paper studies the cardinality-based minimum $s$-$t$ cut problem in hypergraphs, where the cost of each hyperedge depends asymmetrically on the proportion of its vertices on either side of the cut. We first establish that this problem is NP-hard outside the submodular regime—specifically, for $r$-uniform hypergraphs with $r geq 4$, and for all $4$-uniform hypergraphs when $w_2 > 2$. To address non-submodularity, we propose the “optimal projection” strategy: a tight mapping of the non-submodular cost function into the submodular cone, accompanied by a matching approximation algorithm. Furthermore, under the Unique Games Conjecture and the assumption $P eq NP$, we derive tight hardness-of-approximation lower bounds. Our results systematically resolve long-standing gaps in the complexity and approximation theory of non-submodular hypergraph cuts.

This work investigates the lower bound on the number of isolating weight assignments that render the minimum-weight edge unique in inclusion-free hypergraphs. Through combinatorial and hypergraph-theoretic analysis, it establishes that this lower bound is \( n\sum_{j=0}^{d-1} j^{\,n-1} \), and demonstrates that hypergraphs consisting solely of singleton edges precisely attain this bound. The result not only confirms a conjecture by Faber and Harris (2018) but also extends the extremal case of the Isolation Lemma to general objective functions with arbitrary edge offsets, thereby revealing the fundamental influence of hypergraph structure on the complexity of isolating weight assignments.

This work addresses the limited accuracy of upper bound estimation for the Maximum Clique Problem (MCP) by proposing a novel framework that integrates upper bounding functions with graph reduction techniques. The approach introduces adjustable reduction rules—namely $(k,\omega^u)$-core, $(k,\omega^u)$-truss, and the more general $(k,d,\omega^u)$-truss—by embedding upper bound tests into core and truss decompositions for the first time, and further enhances pruning power through struction operations while maintaining computational efficiency. Experimental results demonstrate that the method significantly improves upon multiple classical upper bounds across 73 benchmark graphs. Notably, it reduces certified integer upper bounds to 73, 115, and 168 on the challenging DIMACS instances C500.9, C1000.9, and C2000.9, respectively, and achieves comparable accuracy faster than direct semidefinite programming (SDP) on sparse graphs.

This work investigates combinatorial bounds for codes in general finite metric spaces, with a focus on conditions under which the classical Gilbert–Varshamov (GV) bound can be surpassed. By modeling codes as independent sets in proximity graphs, the authors develop a generalized GV framework applicable to arbitrary metric spaces and introduce novel concepts such as Ramsey–Sidorenko graphs and independence-forcing graphs. Their analysis demonstrates that local subgraph statistics alone are insufficient to exceed the GV bound; instead, global structural properties of the space are essential. Leveraging tools from graph theory, extremal combinatorics, entropy optimization via KKT conditions, and fractional packing techniques, they derive code bounds for both vertex-transitive and non-edge-transitive graphs, establishing density thresholds for several graph families. In particular, they prove that in Hamming spaces, no improvement over the GV bound is possible using only local information, and provide a tight upper bound based on fractional packing.

This work addresses the hypergraph isomorphism problem by proposing IsalHG, a native approach that introduces a closed instruction language to encode finite connected hypergraphs into canonical strings for the first time. It implements a virtual machine based on cyclic doubly linked lists and traversal pointers to parse hypergraph structures. Canonical strings, generated via greedy and backtracking algorithms, serve as complete isomorphism invariants, establishing the first native framework for hypergraph isomorphism testing alongside a benchmark against Levi graph–based baselines (nauty, Traces, and bliss). Experimental validation on 600 instances confirms correctness; however, the method exhibits significantly higher computational cost, running 3–5 orders of magnitude slower than the baselines, with geometric mean slowdowns ranging from 311× to 117,672×.