Link prediction is one of the fundamental problems in computational social science. A particularly common means to predict existence of unobserved links is via structural similarity metrics, such as the number of common neighbors; node pairs with higher similarity are thus deemed more likely to be linked. However, a number of applications of link prediction, such as predicting links in gang or terrorist networks, are adversarial, with another party incentivized to minimize its effectiveness by manipulating observed information about the network. We offer a comprehensive algorithmic investigation of the problem of attacking similarity-based link prediction through link deletion, focusing on two broad classes of such approaches, one which uses only local information about target links, and another which uses global network information. While we show several variations of the general problem to be NP-Hard for both local and global metrics, we exhibit a number of well-motivated special cases which are tractable. Additionally, we provide principled and empirically effective algorithms for the intractable cases, in some cases proving worst-case approximation guarantees.