In the Activation Edge-Multicover problem we are given a multigraph $G=(V,E)$ with activation costs $\{c_{e}^u,c_{e}^v\}$ for every edge $e=uv \in E$, and degree requirements $r=\{r_v:v \in V\}$. The goal is to find an edge subset $J \subseteq E$ of minimum activation cost $\sum_{v \in V}\max\{c_{uv}^v:uv \in J\}$,such that every $v \in V$ has at least $r_v$ neighbors in the graph $(V,J)$. Let $k= \max_{v \in V} r_v$ be the maximum requirement and let $\theta=\max_{e=uv \in E} \frac{\max\{c_e^u,c_e^v\}}{\min\{c_e^u,c_e^v\}}$ be the maximum quotient between the two costs of an edge. For $\theta=1$ the problem admits approximation ratio $O(\log k)$. For $k=1$ it generalizes the Set Cover problem (when $\theta=\infty$), and admits a tight approximation ratio $O(\log n)$. This implies approximation ratio $O(k \log n)$ for general $k$ and $\theta$, and no better approximation ratio was known. We obtain the first logarithmic approximation ratio $O(\log k +\log\min\{\theta,n\})$, that bridges between the two known ratios -- $O(\log k)$ for $\theta=1$ and $O(\log n)$ for $k=1$. This implies approximation ratio $O\left(\log k +\log\min\{\theta,n\}\right) +\beta \cdot (\theta+1)$ for the Activation $k$-Connected Subgraph problem, where $\beta$ is the best known approximation ratio for the ordinary min-cost version of the problem.

翻译：在激活边多重覆盖问题中，给定一个多重图$G=(V,E)$，每条边$e=uv\in E$具有激活成本$\{c_{e}^u,c_{e}^v\}$，以及度数要求$r=\{r_v:v\in V\}$。目标是找到一个边子集$J\subseteq E$，使得在满足每个顶点$v\in V$在图$(V,J)$中至少有$r_v$个邻居的前提下，最小化激活总成本$\sum_{v \in V}\max\{c_{uv}^v:uv \in J\}$。令$k= \max_{v \in V} r_v$为最大度数要求，$\theta=\max_{e=uv \in E} \frac{\max\{c_e^u,c_e^v\}}{\min\{c_e^u,c_e^v\}}$为同一条边两个成本的最大比值。当$\theta=1$时，该问题具有$O(\log k)$的近似比；当$k=1$时，该问题可推广为集合覆盖问题（当$\theta=\infty$时），并具有紧近似比$O(\log n)$。由此可得一般$k$和$\theta$下的近似比$O(k \log n)$，此前未见更优近似比。我们首次获得对数近似比$O(\log k +\log\min\{\theta,n\})$，该结果衔接了已知的两个近似比——$\theta=1$时的$O(\log k)$与$k=1$时的$O(\log n)$。对于激活$k$-连通子图问题，该结果进一步给出近似比$O\left(\log k +\log\min\{\theta,n\}\right) +\beta \cdot (\theta+1)$，其中$\beta$为该问题普通最小成本版本的最佳近似比。