Abstract—Technology mapping and placement have a significant impact on delays in standard cell-based very large scale integrated circuits. Traditionally, these steps are applied separately to optimize the delays, possibly since efficient algorithms that allow the simultaneous exploration of the mapping and placement solution spaces are unknown. In this paper, we present an exact polynomial time algorithm for delay-optimal placement of a tree and extend the same to simultaneous technology mapping and placement for the optimal delay in the tree. We extend the algorithm by employing Lagrangian relaxation technique, which assesses the timing criticality of paths beyond a tree, to optimize the delays in directed acyclic graphs. Experimental results on benchmark circuits in a 70 nm technology show that our algorithms improve timing significantly with remarkably less runtimes compared to a competitive approach of iterative conventional timing-driven mapping and multilevel placement. Index Terms—algorithms, directed acyclic graph, physical synthesis, placement, technology mapping, tree.