While there are well-developed tools for maximizing a  submodular function subject to a matroid constraint , there is much less work on the corresponding supermodular maximization problems. We develop new techniques for attacking these problemsinspired by the continuous greedy method{2}i,j\in AA_{ij}:S\inM, where M is a matroid. These diversity functions have A_{ij}\geq 0 as a measure of dissimilarity of i,j, and A has 0-diagonal. The multilinear framework cannot be directly } 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 . The performance of the adapted algorithm depends on a new smoothness parameter. If F: 0,^2 which measures the approximability of the associated problem \max\{F(x):x\in P\}, for solvable downwards-closed polytopes P. A function F is called one-sided \sigma-smooth if 1{2}u^T\nabla^2 F(x) u\leq\sigma\cdot||u||_1{||x||_1}u^T\nabla F(x) for all u,x\geq 0, x\neq 0. We give an \Omega(} 1 ^n\rightarrow\mathbb{R one-sided \sigma-smooth , then the approximation factor only depends on \sigma. We apply the new algorithm to a broad class of quadratic supermodular functions arising in diversity maximization. This captures metric diversity maximization ( \sigma =2) and negative-type diversity (\sigma=4) . We also develop new methods (inspired by swap rounding and approximate integer decomposition)for rounding quadratics over a matroid polytope. Together with the adapted continuous greedy this leads to a O(\sigma ^{3/2})-approximation . This is the best asymptotic approximation known for this class of quadratics and the evidence suggests that 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. Such set functions are called \gamma-meta-submodular. We develop local search algorithms with approximation factors that depend only on \gamma. We show that the \gamma-meta-submodular families include well-known function classes including meta-submodular functions (\gamma=0), proportionally submodular (\gamma= 1 ), and diversity functions based on negative-type distances or Jensen-Shannon divergence (both \gamma=2) and (semi-)metric diversity)-approximation, independent of n. In this setting, by discretizing, we present a poly-time algorithm for multilinear one-sided \sigma-smooth } functions.
The multilinear framework has achieved the breakthrough 1-1/e approximation for maximizing a monotone submodular function subject to a matroid constraint . This framework has a continuous optimization part and a rounding part. We extend both parts to a wider array of problems. In particular, we make a conceptual contribution by identifying a family of parameterized functions. As a running example we focus on solving diversity problems \max f(S)=\frac{1{2}i,j\in AA_{ij}:S\inM, where M is a matroid. These diversity functions have A_{ij}\geq 0 as a measure of dissimilarity of i,j, and A has 0-diagonal. The multilinear framework cannot be directly } applied to the multilinear extension of such functions. We introduce a new parameter for functions F\in{\bf C^2 which measures the approximability of the associated problem \max\{F(x):x\in P\}, for solvable downwards-closed polytopes P. A function F is called one-sided \sigma-smooth if 1{2}u^T\nabla^2 F(x) u\leq\sigma\cdot||u||_1{||x||_1}u^T\nabla F(x) for all u,x\geq 0, x\neq 0. We give an \Omega(} 1 /\sigma)-approximation for the maximization problem of monotone, normalized one-sided \sigma-smooth F with an additional property: non-positive third order partial derivatives. Using the multilinear framework and new matroid rounding techniques for quadratic objectives, we give an \Omega(1/ \sigma ^{3/2})-approximation for maximizing a \sigma-semi-metric diversity function subject to matroid constraint. This improves upon the previous best bound of \Omega(1/\sigma^2) and we give evidence that it may be tight. For general one-sided smooth functions, we show the continuous process gives an \Omega( 1 /3^{2\sigma)-approximation, independent of n. In this setting, by discretizing, we present a poly-time algorithm for multilinear one-sided \sigma-smooth } functions.
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The multilinear framework has achieved the breakthrough 1-1/e approximation
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, there is much less work on the corresponding supermodular maximization problems. We develop new techniques for attacking these problemsinspired by the continuous greedy method
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multi-linear extension of a submodular function. We first adapt the continuous greedy algorithm to work for general twice-continuously differentiable functions . The performance of the adapted algorithm depends on a new smoothness parameter. If F:
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, then the approximation factor only depends on \sigma. We apply the new algorithm to a broad class of quadratic supermodular functions arising in diversity maximization. This captures metric diversity maximization (
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. This is the best asymptotic approximation known for this class of quadratics and the evidence suggests
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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. Such set functions are called \gamma-meta-submodular. We develop local search algorithms with approximation factors that depend only on \gamma. We show that the \gamma-meta-submodular families include well-known function classes including meta-submodular functions (\gamma=0), proportionally submodular (\gamma=
For general one-sided smooth functions, we show the continuous process gives an \Omega(
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), and diversity functions based on negative-type distances or Jensen-Shannon divergence (both \gamma=2) and (semi-)metric diversity
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