We study an extension of the cardinality-constrained knapsack problem wherein each item has a concave piecewise linear utility structure (CCKP), which is motivated by applications such as resource management problems in monitoring and surveillance tasks. Our main contributions are combinatorial algorithms for the offline CCKP and an online version of the CCKP. 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 property allows us to derive a greedy $(1 - \frac{1}{e})$-approximation algorithm. For the online CCKP in the random order model, we derive a $\frac{10.427}{\alpha}$-competitive algorithm based on $\alpha$-approximation algorithms for the offline CCKP; moreover, we derive stronger guarantees for the cases wherein the cardinality capacity is very small or relatively large. Finally, we investigate the empirical performance of the proposed 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（CCKP），该问题源于监测与监视任务中的资源管理应用。我们的主要贡献在于针对离线CCKP及其在线版本提出组合算法。针对离线问题，我们提出一个完全多项式时间近似方案，并证明该问题可转化为带基数约束的子模函数最大化问题，这一特性使我们能够推导出贪心$(1 - \frac{1}{e})$-近似算法。针对随机顺序模型下的在线CCKP，基于离线CCKP的$\alpha$-近似算法，我们推导出一个$\frac{10.427}{\alpha}$-竞争算法；此外，针对基数容量极小或极大的情形，我们提出了更优的竞争比保证。最后，我们通过数值实验验证了所提算法的实证性能。