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. SIAM J. Comp. 47(5)]. 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. SIAM J. Comp. 47(5)].

中文翻译：

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改进的随机背包模型中背包和GAP在线算法

背包问题是组合优化中的经典问题之一：给定一组项目，每个项目均由其大小和利润指定，目标是在有边界能力的背包中找到最大的利润包装。在在线设置中，一件一件一件的物品被显示出来，如果当前物品被永久包装或丢弃，则必须在抵达后立即且不可撤消地做出决定。我们在随机顺序模型中研究在线变量，其中输入序列是项目集的统一随机排列。我们针对此问题开发了一种随机（1 / 6.65）竞争算法，其性能优于当前最佳竞争比率1 / 8.06 [Kesselheim等。SIAM J.比较 47（5）]。我们的算法基于两个新的见解：我们引入了一种新颖的算法方法，该方法采用了两种给定的算法，针对受限商品类别进行了优化（按输入序列顺序）。此外，我们研究并研究了背包问题与二秘问题的关系。广义分配问题（GAP）除了背包问题外，还包括与调度和匹配有关的几个重要问题。我们显示，在相同的在线环境中，应用建议的顺序方法可产生（1 / 6.99）竞争性GAP随机算法。同样，我们提出的算法优于目前竞争比率1 / 8.06的最佳结果[Kesselheim等。SIAM J.比较 47（5）]。与调度和匹配有关的几个重要问题。我们表明，在相同的在线环境中，应用所提出的顺序方法会产生（1 / 6.99）竞争性GAP随机算法。同样，我们提出的算法优于目前竞争比率1 / 8.06的最佳结果[Kesselheim等。SIAM J.比较 47（5）]。与调度和匹配有关的几个重要问题。我们表明，在相同的在线环境中，应用所提出的顺序方法会产生（1 / 6.99）竞争性GAP随机算法。同样，我们提出的算法优于目前竞争比率1 / 8.06的最佳结果[Kesselheim等。SIAM J.比较 47（5）]。