For the well-known Survivable Network Design Problem (SNDP) we are given an undirected graph $G$ with edge costs, a set $R$ of terminal vertices, and an integer demand $d_{s,t}$ for every terminal pair $s,t\in R$. The task is to compute a subgraph $H$ of $G$ of minimum cost, such that there are at least $d_{s,t}$ disjoint paths between $s$ and $t$ in $H$. If the paths are required to be edge-disjoint we obtain the edge-connectivity variant (EC-SNDP), while internally vertex-disjoint paths result in the vertex-connectivity variant (VC-SNDP). Another important case is the element-connectivity variant (LC-SNDP), where the paths are disjoint on edges and non-terminals. In this work we shed light on the parameterized complexity of the above problems. We consider several natural parameters, which include the solution size $\ell$, the sum of demands $D$, the number of terminals $k$, and the maximum demand $d_\max$. Using simple, elegant arguments, we prove the following results. - We give a complete picture of the parameterized tractability of the three variants w.r.t. parameter $\ell$: both EC-SNDP and LC-SNDP are FPT, while VC-SNDP is W[1]-hard. - We identify some special cases of VC-SNDP that are FPT: * when $d_\max\leq 3$ for parameter $\ell$, * on locally bounded treewidth graphs (e.g., planar graphs) for parameter $\ell$, and * on graphs of treewidth $tw$ for parameter $tw+D$. - The well-known Directed Steiner Tree (DST) problem can be seen as single-source EC-SNDP with $d_\max=1$ on directed graphs, and is FPT parameterized by $k$ [Dreyfus & Wagner 1971]. We show that in contrast, the 2-DST problem, where $d_\max=2$, is W[1]-hard, even when parameterized by $\ell$.
翻译:对于众所周知的幸存者网络设计问题(SNDP),我们得到的是一张无方向的G$G$,加上边价,一个固定的终端网价,对于每个终端对齐美元,t\美元,t\R美元。任务在于计算一个最低成本成本为$H$的子集,这样就有美元,美元和美元之间的脱节路径。如果路径需要是边缘的,我们得到的是精密的 美元(EC-SNDP),而内部的顶端网路则导致顶端连接的变异(VC-SNDP)。另一个重要的例子就是元素连接变异(LC-SNDP),路径在边端和非边端不相交,因此,我们可以看到上述问题的参数复杂性(WD),我们考虑一些自然参数,包括答案大小为$-美元, 平坦的数值,而WD- dral-ralx的数值则是最优的。