Approximating minimum power edge-multi-covers

Abstract

Given an undirected graph with edge costs, the power of a node is the maximum cost of an edge incident to it, and the power of a graph is the sum of the powers of its nodes. Motivated by applications in wireless networks, we consider the following fundamental problem that arises in wireless network design. Given a graph $$G=(V,E)$$G=(V,E) with edge costs and lower degree bounds $$\{r(v):v \in V\}$${r(v):v?V}, the Min-Power Edge-Multicover problem is to find a minimum-power subgraph $$J$$J of $$G$$G such that the degree of every node $$v$$v in $$J$$J is at least $$r(v)$$r(v). Let $$k=\max _{v \in V} r(v)$$k=maxv?Vr(v). For $$k=\Omega (\log n)$$k=Ω(logn), the previous best approximation ratio for the problem was $$O(\log n)$$O(logn), even for uniform costs (Kortsarz et al. 2011). Our main result improves this ratio to $$O(\log k)$$O(logk) for general costs, and to $$O(1)$$O(1) for uniform costs. This also implies ratios $$O(\log k)$$O(logk) for the Min-Power $$k$$k-Outconnected Subgraph and $$O\left( \log k \log \frac{n}{n-k} \right) $$Ologklognn-k for the Min-Power $$k$$k-Connected Subgraph problems; the latter is the currently best known ratio for the min-cost version of the problem when $$n \le k{(k-1)}^2$$n≤k(k-1)2. In addition, for small values of $$k$$k, we improve the previously best ratio $$k+1$$k+1 to $$k+1/2$$k+1/2.