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We propose a new algorithmic framework for traffic-optimal virtual network function (VNF) placement and migration for policy-preserving data centers (PPDCs). As dy- namic virtual machine (VM) traffic must traverse a sequence of VNFs in PPDCs, it generates more network traffic, consumes higher bandwidth, and causes additional traffic delays than a traditional data center. We design optimal, approximation, and heuristic traffic-aware VNF placement and migration algorithms to minimize the total network traffic in the PPDC. In particular, we propose the first traffic-aware constant-factor approximation algorithm for VNF placement, a Pareto-optimal solution for VNF migration, and a suite of efficient dynamic-programming (DP)-based heuristics that further improves the approximation solution. At the core of our framework are two new graph- theoretical problems that have not been studied. Using flow characteristics found in production data centers and realistic traffic patterns, we show that a) our VNF migration techniques are effective in mitigating dynamic traffic in PPDCs, reducing the total traffic cost by up to 73%, b) our VNF placement algorithms yield traffic costs 56% to 64% smaller than those by existing techniques, and c) our VNF migration algorithms outperform the state-of-the-art VM migration algorithms by up to 63% in reducing dynamic network traffic.