We consider the problem of maximizing a fractionally subadditive function under a knapsack constraint that grows over time. An incremental solution to this problem is given by an order in which to include the elements of the ground set, and the competitive ratio of an incremental solution is defined by the worst ratio over all capacities relative to an optimum solution of the corresponding capacity. We present an algorithm that finds an incremental solution of competitive ratio at most $\max\{3.293\sqrt{M},2M\}$, under the assumption that the values of singleton sets are in the range $[1,M]$, and we give a lower bound of $\max\{2.449,M\}$ on the attainable competitive ratio. In addition, we establish that our framework captures potential-based flows between two vertices, and we give a tight bound of~$2$ for the incremental maximization of classical flows with unit capacities.

翻译：我们考虑的是在一个随时间演变的“背包”限制下最大限度地增加一个微小的子相加功能的问题。这个问题的逐步解决是通过一个包括地面组合要素的顺序来实现的,而递增解决方案的竞争性比率则由相对于相应能力的最佳解决办法而言,所有能力的最坏比率来界定。我们提出了一个算法,在假定单吨组合的值在$[$1,M]的范围内,并且我们对可实现的竞争比率给予较低的约束,即$\max$2.449,M$。此外,我们确定我们的框架能够捕捉两个顶端之间的潜在流动,并且我们为具有单位能力的经典流动的递增最大化提供了近2美元的紧凑界限。