A logarithmic approximation algorithm for the activation edge-multicover problem

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∈E, and degree requirements r={r_v:v∈V}. The goal is to find an edge subset J⊆E that minimizes the activation cost ∑_v∈Vmax{c_uv^v:uv∈J}, such that every v ∈ V has at least r_v neighbors in the graph (V, J). Let k=max_v∈Vr_v be the maximum requirement and let [Formula presented] be the maximum quotient between the two costs of an edge. The case θ=1 (when c_e^u=c_e^v for all e=uv∈E) is the well studied MIN-POWER EDGE-MULTICOVER problem, that admits approximation ratio O(log k). On the other hand, for k=1 the problem generalizes the FACILITY LOCATION problem, and admits a tight approximation ratio O(log n). This implies approximation ratio O(klog n) for general k and θ (cf. [2]), and no better approximation ratio was known. Our main result is the first (poly-)logarithmic approximation ratio O(logk+logmin{θ,n}), that bridges between two known approximation ratios – O(log k) for θ=1 and O(log n) for k=1. This also implies approximation ratio O(logk+logmin{θ,n})+β·(θ+1) for the Activation k-Connected Subgraph problem, where β is the best known approximation ratio for the ordinary min-cost version of the problem. We also obtain the following improved approximation ratios for the MIN-POWER EDGE-MULTICOVER problem: (i) k+0.2785 for general costs, improving the ratio of [3] for k ≤ 22. (ii) [Formula presented] for unit costs, improving the ratio 2.16 [3] for k ≤ 10.