  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.
