Submodularity in combinatorial optimization has been a topic of many studies and various algorithmic techniques exploiting submodularity of a studied problem have been proposed. It is therefore natural to ask, in cases where the cost function of the studied problem is not submodular, whether it is possible to approximate this cost function with a proxy submodular function. We answer this question in the negative for two major problems in metric optimization, namely Steiner Tree and Uncapacitated Facility Location. We do so by proving super-constant lower bounds on the submodularity gap for these problems, which are in contrast to the known constant factor cost sharing schemes known for them. Technically, our lower bounds build on strong lower bounds for the online variants of these two problems. Nevertheless, online lower bounds do not always imply submodularity lower bounds. We show that the problem Maximum Bipartite Matching does not exhibit any submodularity gap, despite its online variant being only (1 - 1/e)-competitive in the randomized setting.

翻译：组合优化中的子模性一直是众多研究的主题，并且人们提出了多种利用所研究问题子模性的算法技术。因此，自然要问：当所研究问题的代价函数不满足子模性时，是否可能用代理子模函数来近似该代价函数？我们对此问题给出否定回答，针对度量优化中的两个主要问题——斯坦纳树问题和无容量设施选址问题。我们通过证明这些问题存在超常数下界的子模性缺口来实现这一点，这与已知的常数因子成本分摊方案形成对比。技术上，我们的下界建立在上述两个问题在线变体的强下界之上。然而，在线下界并不总是意味着子模性下界。我们证明，最大二分匹配问题不呈现任何子模性缺口，尽管其在线变体在随机化设置中仅能达到(1 - 1/e)-竞争比。