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A litany of questions from a wide variety of scientific disciplines can be cast as non-monotone submodular maximization problems. Since this class of problems includes max-cut, it is NP-hard. Thus, general purpose algorithms for the class tend to be approximation algorithms. For unconstrained problem instances, one recent innovation in this vein includes an algorithm of Buchbinder et al. (2012) that guarantees a ½ - approximation to the maximum. Building on this, for problems subject to cardinality constraints, Buchbinderet al. (2014) o_er guarantees in the range [0:356; ½ + o(1)]. Earlier work has the best approximation factors for more complex constraints and settings. For constraints that can be characterized as a solvable polytope, Chekuri et al. (2011) provide guarantees. For the online secretary setting, Gupta et al. (2010) provide guarantees. In sum, the current body of work on non-monotone submodular maximization lays strong foundations. However, there remains ample room for future algorithm development.
Jennifer Gillenwater, "Maximization of Non-Monotone Submodular Functions", . January 2014.
Date Posted: 26 February 2014