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.

翻译：《支配H-模式》问题推广了经典的k-支配集问题：对于固定模式H和给定图G，目标是寻找G的诱导子图S，使得（1）S与H同构，且（2）S在G中构成支配集。细粒度复杂性结果表明，在最坏情况输入下，任何相对于朴素暴力算法的显著改进都是不可能的，否则将推翻强指数时间假说。然而，Dransfeld等人（ESA 2025）的最新研究揭示了在稀疏图中尤其存在显著的改进潜力。我们提出：对于特定模式H，具有条件性近似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-模式问题？我们在包含多样化数据集的大规模基准测试中开发并实验评估了多种方法，包括使用格拉斯哥子图求解器（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直接实现与现有求解器性能相当，但我们提出了一种定制的分支定界方法——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性能提升。