We Study Techniques, Approximation Algorithms, Structuralproperties and Lower Bounds Related to Applications of Linear Programs Incombinatorial Optimization. the Following Steiner Tree Problem Is Central: Given a Graph With Adistinguished Subset of Required Vertices, and Costs For Each Edge, Find Aminimum-Cost Subgraph That Connects the Required Vertices. We Also Investigatethe Areas of Network Design, Multicommodity Flows, and Packing/Covering Integerprograms. All of These Problems Are Np-Complete So It Is Natural to Seekapproximation Algorithms With the Best Provable Approximation Ratio. Overall,We Show some New Techniques That Enhance the Already-Substantial Corpus Oflp-Based Approximation Methods, and We Also Look For Limitations of Thesetechniques.