With ever increasing amount of available data on protein-protein interaction (PPI) networks and research revealing that these networks evolve at a modular level, discovery of conserved patterns in these networks becomes an important problem. Recent algorithms on aligning PPI networks target simplified structures such as conserved pathways to render these problems computationally tractable. However, since conserved structures that are parts of functional modules and protein complexes generally correspond to dense subnets of the network, algorithms that are able to extract conserved patterns in terms of general graphs are necessary. With this motivation, we focus here on discovering protein sets that induce subnets that are highly conserved in the interactome of a pair of species. For this purpose, we develop a framework that formally defines the pairwise local alignment problem for PPI networks, models the problem as a graph optimization problem, and presents fast algorithms for this problem. In order to capture the underlying biological processes correctly, we base our framework on duplication/divergence models that focus on understanding the evolution of PPI networks. Experimental results from an implementation of the proposed framework show that our algorithm is able to discover conserved interaction patterns very effectively (in terms of accuracies and computational cost). While we focus on pairwise local alignment of PPI networks in this paper, the proposed algorithm can be easily adapted to finding matches for a subnet query in a database of PPI networks.