Cellular gene expression measurements contain regulatory information that can be used to discover novel network relationships. Here, we present a new algorithm for network reconstruction powered by the adaptive lasso, a theoretically and empirically well-behaved method for selecting the regulatory features of a network. Any algorithms designed for network discovery that make use of directed probabilistic graphs require perturbations, produced by either experiments or naturally occurring genetic variation, to successfully infer unique regulatory relationships from gene expression data. Our approach makes use of appropriately selected cis-expression Quantitative Trait Loci (cis-eQTL), which provide a sufficient set of independent perturbations for maximum network resolution. We compare the performance of our network reconstruction algorithm to four other approaches: the PC-algorithm, QTLnet, the QDG algorithm, and the NEO algorithm, all of which have been used to reconstruct directed networks among phenotypes leveraging QTL. We show that the adaptive lasso can outperform these algorithms for networks of ten genes and ten cis-eQTL, and is competitive with the QDG algorithm for networks with thirty genes and thirty cis-eQTL, with rich topologies and hundreds of samples. Using this novel approach, we identify unique sets of directed relationships in Saccharomyces cerevisiae when analyzing genome-wide gene expression data for an intercross between a wild strain and a lab strain. We recover novel putative network relationships between a tyrosine biosynthesis gene (TYR1), and genes involved in endocytosis (RCY1), the spindle checkpoint (BUB2), sulfonate catabolism (JLP1), and cell-cell communication (PRM7). Our algorithm provides a synthesis of feature selection methods and graphical model theory that has the potential to reveal new directed regulatory relationships from the analysis of population level genetic and gene expression data.