Maximization problems of balancing submodular relevance and supermodular diversity

Relevance and diversity are two desirable properties in data retrieval applications, an important field in data science and machine learning. In this paper, we consider three maximization problems to balance these two factors. The objective function in each problem is the sum of a monotone submodular function f and a supermodular function g, where f and g capture the relevance and diversity of any feasible solution, respectively. In the first problem, we consider a special supermodular diversity function g of a sum-sum format satisfying the relaxed triangle inequality, for which we propose a greedy-type approximation algorithm with an $$\left( 1-1/e,1/(2\alpha )\right) $$ 1 - 1 / e , 1 / ( 2 α ) -bifactor approximation ratio, improving the previous $$\left( 1/(2\alpha ),1/(2\alpha )\right) $$ 1 / ( 2 α ) , 1 / ( 2 α ) -bifactor approximation ratio. In the second problem, we consider an arbitrary supermodular diversity function g, for which we propose a distorted greedy method to give a $$\min \left\{ 1-k_{f}e^{-1},1-k^{g}e^{-(1-k^{g})}\right\} $$ min 1 - k f e - 1 , 1 - k g e - ( 1 - k g ) -approximation algorithm, improving the previous $$k_f^{-1}\left( 1-e^{-k_f(1-k^{g})}\right) $$ k f - 1 1 - e - k f ( 1 - k g ) -approximation ratio, where $$k_f$$ k f and $$k^g$$ k g are the curvatures of the submodular function f and the supermodular funciton g, respectively. In the third problem, we generalize the uniform matroid constraint to the p matroid constraints, for which we present a local search algorithm to improve the previous $$\frac{1-k^g}{(1-k^g)k^f+p}$$ 1 - k g ( 1 - k g ) k f + p -approximation ratio to $$\min \left\{ \frac{p+1-k_f}{p(p+1)},\left( \frac{1-k^g}{p}+\frac{k^g(1-k^g)^2}{p+(1-k^g)^2}\right) \right\} $$ min p + 1 - k f p ( p + 1 ) , 1 - k g p + k g ( 1 - k g ) 2 p + ( 1 - k g ) 2 .