The knapsack problem is one of the classical problems in combinatorial optimization: Given a set of items, each specified by its size and profit, the goal is to find a maximum profit packing into a knapsack of bounded capacity. In the online setting, items are revealed one by one and the decision, if the current item is packed or discarded forever, must be done immediately and irrevocably upon arrival. We study the online variant in the random order model where the input sequence is a uniform random permutation of the item set. We develop a randomized (1/6.65)-competitive algorithm for this problem, outperforming the current best algorithm of competitive ratio 1/8.06 (Kesselheim et al. in SIAM J Comput 47(5):1939–1964, 2018). Our algorithm is based on two new insights: We introduce a novel algorithmic approach that employs two given algorithms, optimized for restricted item classes, sequentially on the input sequence. In addition, we study and exploit the relationship of the knapsack problem to the 2-secretary problem. The generalized assignment problem (GAP) includes, besides the knapsack problem, several important problems related to scheduling and matching. We show that in the same online setting, applying the proposed sequential approach yields a (1/6.99)-competitive randomized algorithm for GAP. Again, our proposed algorithm outperforms the current best result of competitive ratio 1/8.06 (Kesselheim et al. in SIAM J Comput 47(5):1939–1964, 2018).

该背包问题是组合优化中的经典问题之一：给定一组项目，每个项目都由其大小和利润指定，目标是找到一个最大利润包装到容量有限的背包中。在在线设置中，物品被一件一件地显示并且决定，如果当前物品被包装或永远丢弃，必须在到达时立即且不可撤销地做出。我们研究随机顺序模型中的在线变体，其中输入序列是项目集的均匀随机排列。我们为此问题开发了一种随机 (1/6.65) 竞争算法，其性能优于当前竞争比率为 1/8.06 的最佳算法（Kesselheim 等人在 SIAM J Comput 47(5):1939–1964, 2018 中）。我们的算法基于两个新见解：我们引入了一种采用两种给定算法的新算法方法，针对受限制的项目类别进行了优化，在输入序列上按顺序进行。此外，我们研究并利用了背包问题与 2-秘书问题之间的关系。这广义分配问题（GAP）除了背包问题外，还包括几个与调度和匹配相关的重要问题。我们表明，在相同的在线设置中，应用所提出的顺序方法为 GAP 产生了 (1/6.99)-竞争性随机算法。同样，我们提出的算法优于当前竞争比率 1/8.06 的最佳结果（Kesselheim 等人在 SIAM J Comput 47(5):1939–1964, 2018 中）。