A logarithmic approximation algorithm for the activation edge multicover problem (2308.02901v2)

Abstract: 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.