Knapsack Problem With Cardinality Constraint: A Faster FPTAS Through the Lens of Numerical Analysis and Applications

We study the $K$-item knapsack problem (\ie, $1.5$-dimensional knapsack problem), which is a generalization of the famous 0-1 knapsack problem (\ie, $1$-dimensional knapsack problem) in which an upper bound $K$ is imposed on the number of items selected. This problem is of fundamental importance and is known to have a broad range of applications in various fields such as computer science and operation research. It is well known that, there is no FPTAS for the $d$-dimensional knapsack problem when $d\geq 2$, unless P $=$ NP. While the $K$-item knapsack problem is known to admit an FPTAS, the complexity of all existing FPTASs have a high dependency on the cardinality bound $K$ and approximation error $\varepsilon$, which could result in inefficiencies especially when $K$ and $\varepsilon^{-1}$ increase. The current best results are due to \citep{mastrolilli2006hybrid}, in which two schemes are presented exhibiting a space-time tradeoff--one scheme with time complexity $O(n+Kz^{2}/\varepsilon^{2})$ and space complexity $O(n+z^{3}/\varepsilon)$, while another scheme requires a run-time of $O(n+(Kz^{2}+z^{4})/\varepsilon^{2})$ but only needs $O(n+z^{2}/\varepsilon)$ space, where $z=\min\{K,1/\varepsilon\}$. In this paper we close the space-time tradeoff exhibited in \citep{mastrolilli2006hybrid} by designing a new FPTAS with a running time of $O(n)+\widetilde{O}(z^{2}\cdot \max\{K,\varepsilon^{-2}\})$, while simultaneously reaching the $O(n+z^{2}/\varepsilon)$ space complexity. Our scheme provides $\widetilde{O}(\min\{K,\varepsilon^{-2}\})$ and $O(z)$ improvements on the long-established state-of-the-art algorithms in time and space complexity respectively. An salient feature of our algorithm is that it is the \emph{first} FPTAS, which achieves better time and space complexity bounds than the very first standard FPTAS \emph{over all parameter regimes}.