3 research outputs found
Complexity of Discrete Energy Minimization Problems
Discrete energy minimization is widely-used in computer vision and machine
learning for problems such as MAP inference in graphical models. The problem,
in general, is notoriously intractable, and finding the global optimal solution
is known to be NP-hard. However, is it possible to approximate this problem
with a reasonable ratio bound on the solution quality in polynomial time? We
show in this paper that the answer is no. Specifically, we show that general
energy minimization, even in the 2-label pairwise case, and planar energy
minimization with three or more labels are exp-APX-complete. This finding rules
out the existence of any approximation algorithm with a sub-exponential
approximation ratio in the input size for these two problems, including
constant factor approximations. Moreover, we collect and review the
computational complexity of several subclass problems and arrange them on a
complexity scale consisting of three major complexity classes -- PO, APX, and
exp-APX, corresponding to problems that are solvable, approximable, and
inapproximable in polynomial time. Problems in the first two complexity classes
can serve as alternative tractable formulations to the inapproximable ones.
This paper can help vision researchers to select an appropriate model for an
application or guide them in designing new algorithms.Comment: ECCV'16 accepte
Tree-Width and the Computational Complexity of MAP Approximations in Bayesian Networks
The problem of finding the most probable explanation to a designated set of variables given partial evidence (the MAP problem) is a notoriously intractable problem in Bayesian networks, both to compute exactly and to approximate. It is known, both from theoretical considerations and from practical experience, that low tree-width is typically an essential prerequisite to efficient exact computations in Bayesian networks. In this paper we investigate whether the same holds for approximating MAP. We define four notions of approximating MAP (by value, structure, rank, and expectation) and argue that all of them are intractable in general. We prove that efficient value-approximations, structure-approximations, and rank-approximations of MAP instances with high tree-width will violate the Exponential Time Hypothesis. In contrast, we show that MAP can sometimes be efficiently expectation-approximated, even in instances with high tree-width, if the most probable explanation has a high probability. We introduce the complexity class FERT, analogous to the class FPT, to capture this notion of fixed-parameter expectation-approximability. We suggest a road-map to future research that yields fixed-parameter tractable results for expectation-approximate MAP, even in graphs with high tree-width