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In this paper, we propose a new global router, BoxRouter, powered by the concept of box expansion, progressive integer linear programming (PILP), and adaptive maze routing (AMR). BoxRouter first uses a simple prerouting strategy to predict and capture the most congested region with high fidelity as compared to the final routing. Based on progressive box expansion initiated from the most congested region, BoxRouting is performed with PILP and AMR. Our PILP is shown to be much more efficient than the traditional ILP in terms of speed and quality, and the AMR based on multisource multitarget with bridge model is effective in minimizing the congestion and wirelength. It is followed by an effective postrouting step, which reroutes without rip-up to enhance the routing solution further and obtain smooth tradeoff between wirelength and routability. Our experimental results show that the BoxRouter significantly outperforms the state-of-the-art published global routers, e.g., 91 % better routability than Labyrinth (with 14% less wirelength and 3.3times speedup), 79% better routability than Chi-dispersion router (with similar wirelength and 2times speedup), and 4.2% less wirelength and 16times speedup than a multicommodity flow-based router (with similar routability). Additional enhancement in box expansion and postrouting further improves the result with similar wirelength but much better routability than the latest work in global routing.