关于稳定性和赫利性质对支配集问题的影响
On the Impact of Stability and the Helly Property on the Dominating Set Problem
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中文总结 AI 辅助
研究稳定性和赫利性质对支配集问题的影响，通过简单改变去除稳定性要求，在特定图类上得到固定参数可处理算法，推广相关结果且匹配时间复杂度，还能应用于其他问题。
中文摘要 AI 辅助
我们扩展了渐进探索算法框架，该框架为支配集、独立集及其一些变体提供了简单但通用且高效的参数化算法。虽然之前的方法认为稳定性和赫利性质是必要的，但我们表明，对于支配集问题，只需简单改变，就能去除稳定性要求。这在不包含长余匹配或双梯作为半诱导子图的图类上产生了固定参数可处理算法。提升这两个限制之一会使支配集在这些类上成为W[1]难问题。我们的算法推广了关于弱γ闭图的结果以及稀疏理论中的结果，如无处稠密和无双团类。同时，我们在这些类上匹配了先前已知算法的时间复杂度。我们还证明了该技术可轻松应用于距离 - r支配集和集合覆盖问题。
英文摘要
We extend the algorithmic framework of progressive exploration [Fabiański et al., STACS 2019], which yields simple, yet surprisingly general and efficient parameterized algorithms for Dominating Set, Independent Set, and some of their variants. While they identified stability and the Helly property as necessary for their approach, we show that -- with a simple change -- in the case of Dominating Set, one can get rid of the stability requirement. This yields a fixed-parameter tractable algorithm on exactly those graph classes which do not contain long co-matchings or double-ladders as semi-induced subgraphs. Lifting one of these two restrictions makes Dominating Set W[1]-hard on these classes. Our algorithm generalizes results on weakly $γ$-closed graphs, and results from Sparsity theory, e.g., nowhere dense and biclique-free classes. At the same time, we match the time complexity of the previously known algorithms on those classes. We demonstrate that this technique can easily be applied to the Distance-$r$ Dominating Set and the Set Cover problem.
