按秩宽参数化的支配型问题的下界
Lower Bounds for Domination-Type Problems Parameterized by Rank-Width
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中文总结 AI 辅助
本文证明在ETH下，支配集等多种支配型问题不存在2^{o(w²)}n^{O(1)}时间算法，结合已有算法说明秩宽二次依赖是最优的。
中文摘要 AI 辅助
对于秩宽为 \\(w\\) 的图，Bui-Xuan、Telle 和 Vatshelle（《理论计算机科学》，2013）针对固定有限/余有限 \\((\sigma,\rho)\\)-问题的算法，以及 Bergougnoux 和 Kanté（《SIAM 离散数学杂志》，2021）针对连通支配集的算法，运行时间均为 \\(2^{O(w^2)}n^{O(1)}\\)。Bergougnoux、Korhonen 和 Nederlof（STACS 2023）在指数时间假设（ETH）下证明了加权支配集的匹配下界，但未解决无权问题。本文证明：除非 ETH 不成立，否则即使在分裂图、直径至多为4的二分图上，即使提供了秩分解或见证顶点序，支配集也不存在 \\(2^{o(w^2)}n^{O(1)}\\) 时间的算法。该证明用双护卫小工具替代早期的权重，利用低秩等式小工具通过秩为 \\(O(k)\\) 的割传递 \\(k^2\\) 个分配比特。该构造还对受限图类上的独立支配集、连通支配集和全支配集给出相同下界，且适用于广泛的 \\((\sigma,\rho)\\)-集问题族，包括 \\(\sigma\\) 既非有限也非余有限的情况，以及整个非平凡余有限-余有限极小化 regime。目标预算内的每个解都有目标大小，且与满足赋值一一对应。因此，在计数指数时间假设（\\(\\#\text{ETH}\\)）下，对于大小至多为或恰好为目标的解的计数，也成立相同下界。结合已知算法，本文结果表明，对于上述经典问题及所覆盖的全部有限/余有限 regime，秩宽 \\(w\\) 的二次依赖在指数的常数因子范围内是最优的。
英文摘要
For graphs of rank-width \(w\), the algorithms of Bui-Xuan, Telle, and Vatshelle (\emph{Theor. Comput. Sci.}, 2013) for fixed finite/cofinite \((σ,ρ)\)-problems and of Bergougnoux and Kanté (\emph{SIAM J. Discrete Math.}, 2021) for Connected Dominating Set run in \(2^{O(w^2)}n^{O(1)}\) time. Bergougnoux, Korhonen, and Nederlof (STACS 2023) proved a matching lower bound under the Exponential Time Hypothesis (ETH) for \emph{Weighted} Dominating Set, but left the unweighted problem open. We prove that, unless ETH fails, Dominating Set admits no \(2^{o(w^2)}n^{O(1)}\)-time algorithm, even on split graphs and, separately, on bipartite graphs of diameter at most four, and even with a rank-decomposition or witnessing vertex order supplied. The proof replaces the earlier weights by a two-guard gadget and uses a low-rank equality gadget to carry \(k^2\) assignment bits through cuts of rank \(O(k)\). The construction also gives the same lower bound for Independent, Connected, and Total Dominating Set on restricted graph classes and applies to a broad family of \((σ,ρ)\)-set problems. This family includes cases in which \(σ\) is neither finite nor cofinite and contains the entire nontrivial cofinite--cofinite minimization regime. Every solution within the target budget has target size and corresponds bijectively to a satisfying assignment. Under the counting Exponential Time Hypothesis (\(\#\mathrm{ETH}\)), the same bounds therefore hold for counting solutions of size at most or exactly the target. Together with the known algorithms, our results show that the quadratic dependence on the rank-width \(w\) is optimal up to constant factors in the exponent for the classical problems above and throughout the covered finite/cofinite regime.
