We study an extension of the cardinality-constrained knapsack problem where each item has a concave piecewise-linear utility structure. Our main contributions are approximation algorithms for the problem and the investigation of an online version in the random order model. For the offline problem, we present a fully polynomial-time approximation scheme and show that it can be cast as the maximization of a submodular function with cardinality constraints; the latter result allows us to derive a greedy $(1 - \frac{1}{e})$-approximation algorithm. For the online problem in the random order model, we present a 6.027-competitive algorithm. Finally, we investigate the empirical performance of the greedy and online algorithms in numerical experiments.

翻译：我们研究了基数约束背包问题的一个扩展，其中每个物品具有凹分段线性效用结构。本文的主要贡献在于为该问题设计了近似算法，并考察了随机顺序模型下的在线版本。针对离线问题，我们提出了一个完全多项式时间近似方案，并证明该问题可转化为基数约束下的子模函数最大化问题；这一结果使我们能够推导出一个贪心$(1 - \frac{1}{e})$近似算法。针对随机顺序模型下的在线问题，我们提出了一个6.027竞争比算法。最后，我们通过数值实验评估了贪心算法与在线算法的实际性能。