We study theoretical runtime guarantees for a class of optimization problems that occur in a wide variety of inference problems. these problems are motivated by the lasso framework and have applications in machine learning and computer vision. Our work shows a close connection between these problems and core questions in algorithmic graph theory. While this connection demonstrates the difficulties of obtaining runtime guarantees, it also suggests an approach of using techniques originally developed for graph algorithms. We then show that most of these problems can be formulated as a grouped least squares problem, and give efficient algorithms for this formulation. Our algorithms rely on routines for solving quadratic minimization problems, which in turn are equivalent to solving linear systems. Finally we present some experimental results on applying our approximation algorithm to image processing problems.