发表机构
The Institute of Mathematical Sciences, HBNI; University of Bergen（数学科学研究所，HBNI; 卑尔根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对半阶梯无关图上的连通支配集问题，提出固定参数算法与近似核化框架，利用极小集合覆盖的紧致表示和分组支配核，实现多项式时间有损压缩。
AI 中文摘要
我们研究了闭邻域集合系统为$d$-半阶梯无关的图上的\textsc{连通支配集}问题。这一结构条件严格推广了双团无关的设定，并为连通性约束的支配问题提供了一个自然的研究框架。我们针对该问题在该图类上同时获得了一个固定参数算法和一个近似核化框架。我们的算法结果基于一个新的紧致表示定理，该定理适用于$d$-半阶梯无关集合系统中包含意义下的极小集合覆盖。尽管大小为至多$k$的极小集合覆盖的数量可能多达$n^{\Omega(k)}$，我们证明了所有这些集合覆盖仍然可以被一个至多包含$k^{kd+1}$个元组的族所编码，并且该族可以在时间$\Oh(k^{kd+2}\cdot nm)$内被枚举。将该表示与\textsc{组斯坦纳树}子程序相结合，我们获得了\textsc{连通集合覆盖}的一个算法，该算法进而产生了\textsc{连通支配集}的一个算法，其运行时间为$k^{kd+2}\cdot 2^k \cdot n^{\Oh(1)}$，并且使用多项式空间。对于预处理结果，我们引入了分组支配核和支配者核，并证明了它们在$d$-半阶梯无关图中的大小具有多项式上界。利用这些结构，对于每个固定的$d$和$\varepsilon>0$，我们获得了\textsc{连通支配集}到规模为$k^{\Oh(d^2/\varepsilon)}$的等价简化实例的多项式时间$(1+\varepsilon)$-有损压缩。该简化实例是$(d+2)$-半阶梯无关图上的\textsc{连通支配集}实例。
英文摘要
We study \textsc{Connected Dominating Set} on graphs whose closed-neighborhood set systems are $d$-semi-ladder-free. This structural condition strictly generalizes the biclique-free setting and provides a natural regime for connectivity-constrained domination. We obtain both a fixed-parameter algorithm and an approximate kernelization framework for the problem on this class. Our algorithmic result is based on a new compact representation theorem for inclusion-wise minimal set covers in $d$-semi-ladder-free set systems. Although the number of minimal set covers of size at most $k$ may be as large as $n^{Ω(k)}$, we show that all such set covers can nevertheless be encoded by a family of at most $k^{kd+1}$ tuples, and that this family can be enumerated in time $\Oh(k^{kd+2}\cdot nm)$. Combining this representation with a \textsc{Group Steiner Tree} subroutine, we obtain an algorithm for \textsc{Connected Set Cover}, which in turn yields an algorithm for \textsc{Connected Dominating Set} running in time $k^{kd+2}\cdot 2^k \cdot n^{\Oh(1)}$ and polynomial space. For the preprocessing result, we introduce grouped domination cores and dominator cores, and prove polynomial upper bounds on their sizes in $d$-semi-ladder-free graphs. Using these structures, we obtain, for every fixed $d$ and $\varepsilon>0$, a polynomial-time $(1+\varepsilon)$-lossy compression for \textsc{Connected Dominating Set} to an equivalent reduced instance of size $k^{\Oh(d^2/\varepsilon)}$. The reduced instance is a \textsc{Connected Dominating Set} instance on a $(d+2)$-semi-ladder-free graph.
CommentsThis is an archived version of the paper accepted at ISAAC 2026
