Summary: There has been much recent algorithmic work on the problem of reconstructing the evolutionary history of biological species. Computer virus specialists are interested in finding the evolutionary history of computer viruses – a virus is often written using code fragments from one or more other viruses, which are its immediate ancestors. A phylogeny for a collection of computer viruses is a directed acyclic graph whose nodes are the viruses and whose edges map ancestors to descendants and satisfy the property that each code fragment is “invented” only once. To provide a simple explanation for the data, we consider the problem of constructing such a phylogeny with a minimum number of edges. This optimization problem is NP-hard, and we present positive and negative results for associated approximation problems. When tree solutions exist, they can be constructed and randomly sampled in polynomial time.