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A major challenge in molecular systems biology is to understand how proteins work to transmit external signals to changes in gene expression. Computationally reconstructing these signaling pathways from protein interaction networks can help understand what is missing from existing pathway databases. We formulate a new pathway reconstruction problem, one that iteratively grows directed acyclic graphs (DAGs) from a set of starting proteins in a protein interaction network. We present an algorithm that provably returns the optimal DAGs for two different cost functions and evaluate the pathway reconstructions when applied to six diverse signaling pathways from the NetPath database. The optimal DAGs outperform an existing k-shortest paths method for pathway reconstruction, and the new reconstructions are enriched for different biological processes. Growing DAGs is a promising step toward reconstructing pathways that provably optimize a specific cost function.