工程化支配模式：一个细粒度案例研究
Engineering Dominating Patterns: A Fine-grained Case Study
AI总结：
针对支配H-模式问题，本文提出一种结合精细剪枝的定制分支定界方法，在多种数据集上实现了比现有求解器高达两个数量级的性能改进。
摘要：
支配 H-模式（Dominating H-Pattern）问题推广了经典的 k-支配集（k-Dominating Set）问题：对于固定的模式 H 和给定的图 G，目标是找到 G 的一个导出子图 S，使得 (1) S 与 H 同构，且 (2) S 在 G 中构成支配集。细粒度复杂度结果表明，在最坏情况输入下，对朴素的暴力算法进行任何显著改进都是不太可能的，因为这将反驳强指数时间假设。尽管如此，Dransfeld 等人最近的一项工作（ESA 2025）揭示了显著的改进潜力，特别是在稀疏图中。我们提出疑问：对于选定的模式 H，具有条件性近乎最优最坏情况性能的算法能否在实际输入上高效求解支配 H-模式？我们在包含多种数据集的大型基准上开发并实验评估了多种方法，包括使用 Glasgow 子图求解器（GSS）、SAT 求解器 Kissat 和 ILP 求解器 Gurobi 的基线方法。值得注意的是，虽然具有条件性接近最优最坏情况保证的算法的直接实现表现与现有求解器相当，但我们提出了一种定制的分支定界（Branch-&-Bound）方法，辅以精细的剪枝技术，在我们的测试实例上实现了高达两个数量级的改进。
英文摘要：
The \emph{Dominating $H$-Pattern} problem generalizes the classical $k$-Dominating Set problem: for a fixed \emph{pattern} $H$ and a given graph $G$, the goal is to find an induced subgraph $S$ of $G$ such that (1) $S$ is isomorphic to $H$, and (2) $S$ forms a dominating set in $G$. Fine-grained complexity results show that on worst-case inputs, any significant improvement over the naive brute-force algorithm is unlikely, as this would refute the Strong Exponential Time Hypothesis. Nevertheless, a recent work by Dransfeld et al. (ESA 2025) reveals some significant improvement potential particularly in \emph{sparse} graphs. We ask: Can algorithms with conditionally almost-optimal worst-case performance solve the Dominating $H$-Pattern, for selected patterns $H$, efficiently on practical inputs? We develop and experimentally evaluate several approaches on a large benchmark of diverse datasets, including baseline approaches using the Glasgow Subgraph Solver (GSS), the SAT solver Kissat, and the ILP solver Gurobi. Notably, while a straightforward implementation of the algorithms -- with conditionally close-to-optimal worst-case guarantee -- performs comparably to existing solvers, we propose a tailored Branch-\&-Bound approach -- supplemented with careful pruning techniques -- that achieves improvements of up to two orders of magnitude on our test instances.
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