text: We design new approximation algorithms for the problems of optimizing submodular and supermodular functions subject to a single matroid constraint. Specifically, we consider the case in which we wish to maximize a nondecreasing submodular function or minimize a nonincreasing supermodular function in the setting of bounded total curvature @math . In the case of submodular maximization with curvature @math , we obtain a @math -approximation --- the first improvement over the greedy @math -approximation of Conforti and Cornuejols from 1984, which holds for a cardinality constraint, as well as recent approaches that hold for an arbitrary matroid constraint. Our approach is based on modifications of the continuous greedy algorithm and non-oblivious local search, and allows us to approximately maximize the sum of a nonnegative, nondecreasing submodular function and a (possibly negative) linear function. We show how to reduce both submodular maximization and supermodular minimization to this general problem when the objective function has bounded total curvature. We prove that the approximation results we obtain are the best possible in the value oracle model, even in the case of a cardinality constraint. We define an extension of the notion of curvature to general monotone set functions and show @math -approximation for maximization and @math -approximation for minimization cases. Finally, we give two concrete applications of our results in the settings of maximum entropy sampling, and the column-subset selection problem.
  @cite_25: We consider the problem of minimizing a supermodular set function whose special case is the well-known NP-hard p-median problem. The main result of the paper is a tight bound on the approximation ratio of a greedy heuristic (discrete analog of the steepest descent algorithm) for this problem.
summary: We shall also consider the problem of minimizing nonincreasing functions @math . By analogy with total curvature, Il'ev @cite_25 defines the @math of a nonincreasing supermodular function. His definition, which is stated in terms of the marginal of the function, is equivalent to when reformulated in terms of marginal gains. He showed that, in contrast to submodular maximization, the simple greedy heuristic does not give a constant factor approximation algorithm in the general case. However, when the supermodular function @math has total curvature at most @math , he shows that the reverse greedy algorithm is an @math -approximation where @math .