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 $\max\{2.618,M\}$ on the attainable competitive ratio. In addition, we establish that our framework captures potential-based flows between two vertices, and we give a lower bound of $\max\{2,M\}$ and an upper bound of $2M$ for the incremental maximization of classical flows with capacities in $[1,M]$ which is tight for the unit capacity case.

翻译：我们考虑在随时间增长的背包约束下最大化分数次可加函数的问题。该问题的一个增量解由地面集合中元素的包含顺序给出，而增量解的竞争比定义为所有容量下相对于对应容量最优解的最差比率。我们提出一种算法，在单元素集合的值位于区间$[1,M]$的假设下，该算法能找到竞争比至多为$\max\{3.293\sqrt{M},2M\}$的增量解，并对可实现的竞争比给出了$\max\{2.618,M\}$的下界。此外，我们证明该框架能描述两顶点间的势流，并对容量在$[1,M]$内的经典流增量最大化问题给出下界$\max\{2,M\}$和上界$2M$，该上界在单位容量情况下是紧的。