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While there are well-developed tools for maximizing a submodular function f (S) subject to a matroid constraint S ∈ M, there is much less work on the corresponding supermodular maximization problems. We develop new techniques for attacking these problems inspired by the continuous greedy method applied to the multi-linear extension of a submodular function. We first adapt the continuous greedy algorithm to work for general twice-continuously differentiable functions. Reminiscent of how the Lipschitz constant governs the convergence rate in convex optimization, the performance of the adapted algorithm depends on a new smoothness parameter. If F : [0, 1] n → R ≥0 is one-sided σ-smooth, then it yields an approximation factor depending only on σ. We apply the new algorithm to a broad class of quadratic supermodular functions arising in diversity maximization. The case σ = 2 captures metric diversity maximization and general σ includes the densest subgraph problem. We also develop new methods for rounding quadratics over a matroid polytope. These are based on extensions to swap rounding and approximate integer decomposition. Together with the adapted continuous greedy this leads to a O(σ 3/2 )-approximation. This is the best asymptotic approximation known for this class of diversity maximization and we give some evidence for why we believe it may be tight.We then consider general (non-quadratic) functions. We give a broad parameterized family of monotone functions which include submodular functions and the just-discussed supermodular family of discrete quadratics. The new family is defined by restricting the one-sided smoothness condition to the boolean hypercube; such set functions are called γ-meta-submodular. We develop local search algorithms with approximation factors that depend only on γ. We show that the γ-meta-submodular families include well-known function classes including meta-submodular functions (γ = 0), proportionally submodular (γ = 1), and diversity functions based on negativetype distances or Jensen-Shannon divergence (both γ = 2) and (semi-)metric diversity functions.

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A common approach to solve a combinatorial optimization problem is to first solve a continous relaxation and then round the fractional solution. For the latter, the framework of contention resolution schemes (or CR schemes) introduced by Chekuri, Vondrak, and Zenklusen, has become a general and successful tool. A CR scheme takes a fractional point x in a relaxation polytope, rounds each coordinate xi independently to get a possibly non-feasible set, and then drops some elements in order to satisfy the independence constraints. Intuitively, a CR scheme is c-balanced if every element i is selected with probability at least c • xi.It is known that general matroids admit a (1 − 1/e)-balanced CR scheme, and that this is (asymptotically) optimal. This is in particular true for the special case of uniform matroids of rank one. In this work, we provide a simple monotone CR scheme with a balancedness factor of 1 − e −k k k /k! for uniform matroids of rank k, and show that this is optimal. This result generalizes the 1 − 1/e optimal factor for the rank one (i.e. k = 1) case, and improves it for any k > 1. Moreover, this scheme generalizes into an optimal CR scheme for partition matroids.