The fractional knapsack problem is one of the classical problems in combinatorial optimization, which is well understood in the offline setting. However, the corresponding online setting has been handled only briefly in the theoretical computer science literature so far, although it appears in several applications. Even the previously best known guarantee for the competitive ratio was worse than the best known for the integral problem in the popular random order model. We show that there is an algorithm for the online fractional knapsack problem that admits a competitive ratio of 4.39. Our result significantly improves over the previously best known competitive ratio of 9.37 and surpasses the current best 6.65-competitive algorithm for the integral case. Moreover, our algorithm is deterministic in contrast to the randomized algorithms achieving the results mentioned above.

中文翻译：

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随机顺序模型中分数背包的改进在线算法

分数背包问题是组合优化中的经典问题之一，在离线设置中很好理解。然而，到目前为止，相应的在线设置仅在理论计算机科学文献中得到了简要处理，尽管它出现在多个应用程序中。即使是以前最著名的竞争比率保证也比流行的随机顺序模型中最著名的积分问题更糟糕。我们展示了在线分数背包问题的算法，该算法允许竞争率为 4.39。我们的结果显着改善了之前最著名的 9.37 竞争比率，并超过了当前最好的 6.65 竞争算法积分情况。而且，