Packing disjoint subgraphs in a given graph is a fundamental problem with many applications. Motivated by political districting, we focus on connected subgraphs that are compact (e.g., having constant radius from a single center vertex) and that satisfy additional composition requirements, such as a minimum population/weight threshold or balanced weight types (e.g., political affiliations). We aim to maximize coverage by disjoint districts that meet these requirements.
In this work, we present new results that substantially improve the previously known bounds on balanced star districts for planar and minor-free graphs (Dharangutte et al. 2025). In particular, we improve the approximation factor from $O(\log n)$ to $O(1)$ for packing balanced star districts using the exact same algorithm, but with a refined analysis. We also extend the results beyond planar graphs to minor-free graphs and an even broader family of graphs of bounded expansion. Additionally, we obtain an $O(1)$ approximation for packing radius-$k$ districts (with a constant $k$) in planar and apex-minor-free graphs. In order to get a $(1+\varepsilon)$ approximation on the max coverage, we show that this can be achieved if we allow a slight relaxation of the balancedness parameters (by a factor that can be made arbitrarily close to $1$), for bounded radius-$k$ districts on planar and apex-minor-free graphs.
We show that all of these results can also be obtained if we enforce a minimum weight threshold for each district as the composition requirement, rather than balancedness. We present various results on hardness and hardness of approximation for this variant, by graph and district types.
Connected Domatic Packings in Node-capacitated Graphs
一个图中的顶点集是一个支配集，如果这个集外的每个顶点都有一个邻居在这个集中。一个支配集是连通的，如果它的顶点诱导的子图是连通的。连通域划分问题要求将节点划分为连通支配集。图的连通域数是最大连通域划分的大小，它是一个研究得很好的图参数，在无线网络设计中有应用。本文研究了\emph{node-capacitated}图中连通域划分问题的分数阶对应问题。令$n$为图中节点的数量，令$k$为$G$中节点分隔符的最小容量。我们的算法构造的填充集的大小与$k$成正比，并且我们的算法最多可以填充$k$连通控制集，这取决于节点的容量.我们的一些主要贡献如下：\begin{itemize} \item An algorithm for constructing a fractional connected domatic packing of size $Ω(k)$ for node-capacitated planar and minor-closed families of graphs. \item An algorithm for constructing a fractional connected domatic packing of size $Ω(k / \ln{n})$ for node-capacitated general graphs. \end{itemize}
An $O(\log OPT)$-approximation for covering and packing minor models of $θ_r$
给定两个图$G$和$H$，我们定义了$\textsf{v-cover}_{H}(G)$（分别为. $\textsf{e-cover}_{H}(G)$）作为顶点的最小数目（分别边），其从$G$中移除产生没有与${H}$同构的任何次同构的图。也称为$\textsf{v-pack}_{H}(G)$（分别$\textsf{v-pack}_{H}(G)$）是顶点的最大数目-（相应地，$G$的边）不相交子图，其中包含与$H$同构的次同构。我们用$θ_r$表示两个顶点之间有PDMATHAMT平行边的图。当$H=θ_r$时，参数$\textsf{v-cover}_{H}$、$\textsf{e-cover}_{H}$、$\textsf{v-pack}_{H}$和$\textsf{v-pack}_{H}$是NP难计算的（对于足够大的$r$值）。利用[Minors in graphs of large $θ_r$ -girth，Chatzidimitriou等人，arXiv：1510.03041]，我们给出了一个算法证明，如果$\textsf{v-pack}_{θ_r}(G)\leq k$，那么$\textsf{v-cover}_{θ_r}(G) = O(k\log k)$，并且类似地对于$\textsf{v-pack}_{θ_r}$和$\textsf{e-cover}_{θ_r}$。换句话说，包含$…
