We study a variety of NP-Complete network connectivity problems. Our primary results come from a novel Dual-Based approach to approximating network design problems with cut-based linear programming relaxations. This approach gives a 3/2-approximation to Minimum 2-Edge-Connected Spanning Subgraph that is equivalent to a previously proposed algorithm. One well-studied branch of network design models ad hoc networks where each node can either operate at high or low power. If we allow unidirectional links, we can formalize this into the problem Dual Power Assignment (DPA). Our Dual-Based approach gives a 3/2-approximation to DPA, improving the previous best known approximation of 11/7 ≈ 1.57.;Another standard network design problem is Minimum Strongly Connected Spanning Subgraph (MSCS). We propose a new problem generalizing MSCS and DPA called Star Strong Connectivity (SSC). Then we show that our Dual-Based approach achieves a 1.6-approximation ratio on SSC. As a result of our Dual-Based approximations, we prove new upper bounds on the integrality gaps of these problems. For completeness, we present a family of instances of MSCS (and thus SSC) with integrality gap approaching 4/3.
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