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Then, we provide a general graph-theoretic framework to decouple scale, rotation, and translation estimation, which allows solving in cascade for the three transformations. Despite the fact that each subproblem is still non-convex and combinatorial in nature, we show that (i) TLS scale and (component-wise) translation estimation can be solved in polynomial time via adaptive voting, (ii) TLS rotation estimation can be relaxed to a semidefinite program (SDP) and the relaxation is tight, even in the presence of extreme outlier rates, and (iii) the graph-theoretic framework allows drastic pruning of outliers by finding the maximum clique. We name the resulting algorithm TEASER (Truncated least squares Estimation And SEmidefinite Relaxation). While solving large SDP relaxations is typically slow, we develop a second fast and certifiable algorithm, named TEASER++, that uses graduated non-convexity to solve the rotation subproblem and leverages Douglas-Rachford Splitting to efficiently certify global optimality. For both algorithms, we provide theoretical bounds on the estimation errors, which are the first of their kind for robust registration problems. 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