We consider a class of submodular maximization problems in which decision-makers have limited access to the objective function. We explore scenarios where the decision-maker can observe only pairwise information, i.e., can evaluate the objective function on sets of size two. We begin with a negative result that no algorithm using only $k$-wise information can guarantee performance better than $k/n$. We present two algorithms that utilize only pairwise information about the function and characterize their performance relative to the optimal, which depends on the curvature of the submodular function. Additionally, if the submodular function possess a property called supermodularity of conditioning, then we can provide a method to bound the performance based purely on pairwise information. The proposed algorithms offer significant computational speedups over a traditional greedy strategy. A by-product of our study is the introduction of two new notions of curvature, the $k$-Marginal Curvature and the $k$-Cardinality Curvature. Finally, we present experiments highlighting the performance of our proposed algorithms in terms of approximation and time complexity.
翻译:我们考虑的是一类亚模式最大化问题,即决策者对客观功能的接触有限。我们探讨了决策者只能对称信息进行观察的情景,即对称信息,可以对二号数组的客观功能进行评估。我们首先得出一个负面结果,即任何仅使用美元/美元信息算法的算法都不能保证业绩优于美元/美元。我们提出两种算法,这些算法仅使用关于该功能的对称信息,并描述其相对于最佳功能的性能特征,这取决于子模式函数的曲调。此外,如果子模式函数拥有所谓的超模调特性,那么我们就可以提供一种纯粹基于对称信息的方法来约束该功能。提议的算法为传统的贪婪战略提供了重大的计算超速。我们研究的副产品是引入两种新的曲线概念,即美元-马尔基纳尔曲线和美元-卡尔迪纳利曲线。最后,我们提出实验,以近似和时间复杂性的方式突出我们提议的算法的性能。