A connected partition is a partition of the vertices of a graph into sets that induce connected subgraphs. Such partitions naturally occur in many application areas such as road networks, and image processing. We consider Balanced Connected Partitions (BCP), where the two classical objectives for BCP are to maximize the weight of the smallest, or minimize the weight of the largest component. We study BCP on c-claw-free graphs, the class of graphs that do not have $K_{1,c}$ as an induced subgraph, and present efficient (c-1)-approximation algorithms for both objectives. In particular, due to the (3-)claw-freeness of line graphs, this also implies a 2-approximations for the edge-partition version of BCP in general graphs. In the 1970s Gy\H{o}ri and Lov\'{a}sz showed for natural numbers $w_1,\dots,w_k$ where $\sum_i w_i$ is the vertex size, that if $G$ is k-connected, then there exist a connected k-partition with part sizes $w_1,\dots,w_k$. However, to this day no polynomial algorithm to compute such partitions exists for k>4. Towards finding such a partition $T_1,\dots, T_k$, we show how to efficiently compute connected partitions that at least approximately meet the target weights, subject to the mild assumption that each $w_i$ is greater than the weight of the heaviest vertex. In particular, we give a 3-approximation for both the lower and the upper bounded version i.e. we guarantee that each $T_i$ has weight at least $\frac{w_i}{3}$ or that each $T_i$ has weight most $3w_i$, respectively. Also, we present a both-side bounded version that produces a connected partition where each $T_i$ has size at least $\frac{w_i}{3}$ and at most $\max(\{r,3\}) w_i$, where $r \geq 1$ is the ratio between the largest and smallest value in $w_1, \dots, w_k$. In particular for the balanced version, i.e.~$w_1=w_2=, \dots,=w_k$, this gives a partition with $\frac{1}{3}w_i \leq w(T_i) \leq 3w_i$.
翻译:连接的分区是 3 个图形的顶端 3 个图表的分区 3 个图形的顶端 3 个 3 个 3 个 美元 3 美元 作为 连接的子集 。 这种顶部自然出现在许多最小应用区域, 如道路网络和图像处理 。 我们考虑平衡连接分区 (BCP), 其中BCP的两个经典目标是最大限度地增加最小部分的重量, 或将最大部分的重量最小部分。 在 1970s Gy\H{o}ri 和 Lov\\ {a}z 显示自然数字 $w 1, 美元, 美元作为引导的子节, 并展示效率(c-1) 美元 3 美元 美元 和 最高比例值 3 。 当 美元 和 美元 美元(xxxxxxx) 之间, 美元比 最低值的顶值 3 美元 。