In this paper we describe exact and heuristic algorithms for the comparison of 3D structures via their contact maps. Given two contact maps, we consider the problem of finding the optimal sequence — order dependent and sequence — order independent alignments. We describe an integer programming formulation of the problems along with a Lagrangian relaxation yielding stronger bounds than previous approaches. This relaxation is used to drive both a greedy heuristic capable of finding near — optimal solutions and an exact branch — and — bound algorithm. We report computational results for an application of our model to the problem of aligning 3D structures of folded proteins from the Protein Data Bank.