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 \(\left( 1-1/e,1/(2\alpha )\right) \)-bifactor approximation ratio, improving the previous \(\left( 1/(2\alpha ),1/(2\alpha )\right) \)-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 \(\min \left\{ 1-k_{f}e^{-1},1-k^{g}e^{-(1-k^{g})}\right\} \)-approximation algorithm, improving the previous \(k_f^{-1}\left( 1-e^{-k_f(1-k^{g})}\right) \)-approximation ratio, where \(k_f\) and \(k^g\) 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 \(\frac{1-k^g}{(1-k^g)k^f+p}\)-approximation ratio to \(\min \left\{ \frac{p+1-k_f}{p(p+1)},\left( \frac{1-k^g}{p}+\frac{k^g(1-k^g)^2}{p+(1-k^g)^2}\right) \right\} \).

相关性和多样性是数据检索应用程序中的两个理想属性，这是数据科学和机器学习的一个重要领域。在本文中，我们考虑了三个最大化问题来平衡这两个因素。每个问题中的目标函数是单调子模函数f和超模函数g的总和，其中f和g 分别捕获任何可行解的相关性和多样性。在第一个问题中，我们考虑满足松弛三角不等式的和-和格式的特殊超模分集函数g，为此我们提出了一种贪婪型逼近算法，其中\(\left( 1-1/e,1/ (2\alpha )\right) \)-bifactor 逼近比，提高了之前的\(\left( 1/(2\alpha ),1/(2\alpha )\right) \) -bifactor 逼近比。在第二个问题中，我们考虑一个任意的超模分集函数g，为此我们提出了一种扭曲的贪婪方法来给出一个\(\min \left\{ 1-k_{f}e^{-1},1-k^ {g}e^{-(1-k^{g})}\right\} \) -近似算法，改进了之前的\(k_f^{-1}\left( 1-e^{-k_f(1 -k ^ {G}）} \右）\） -近似比率，其中\（k_f \）和\（K ^克\）是子模函数的曲率˚F和超模功能可按克， 分别。在第三个问题中，我们将一致拟阵约束推广到p 个拟阵约束，为此我们提出了一种局部搜索算法来改进之前的\(\frac{1-k^g}{(1-k^g)k^ f+p}\) - 近似比\(\min \left\{ \frac{p+1-k_f}{p(p+1)},\left( \frac{1-k^g}{p }+\frac{k^g(1-k^g)^2}{p+(1-k^g)^2}\right) \right\} \)。