A tight 1/3-approximation algorithm and fully polynomial-time approximation schemes for the Colored Knapsack Problem

2026-09-15

AI Summary

该研究针对彩色背包问题，提出了一种1/3近似算法和两种完全多项式时间近似方案，通过线性规划松弛和利润缩放改进了原有动态规划算法的效率。

Abstract

The $\textit{Colored Knapsack Problem}$ (ColKP) generalizes the classical Knapsack Problem by partitioning the items into color classes and requiring the selected items to admit an ordering in which consecutive items have different colors. The problem is weakly $\mathcal{NP}$-hard and admits two pseudo-polynomial dynamic programming (DP) algorithms proposed in the literature. These two DP algorithms have worst-case running times $O(b \, n^4)$ and $O(b^2 \, n^3)$, respectively, where $b$ is the knapsack capacity and $n$ is the number of items. We develop the first approximation algorithm for the ColKP. By rounding an optimal basic solution of the linear programming relaxation of its natural integer programming formulation and repairing color feasibility, we obtain a linear-time approximation-algorithm whose worst-case performance ratio is $1/3$. We then reformulate both DP algorithms so that profit, rather than knapsack capacity, indexes their pseudo-polynomial dimension, and combine them with profit scaling to obtain two fully polynomial-time approximation schemes (FPTASs). The first FPTAS runs in $O(n^5/\varepsilon)$ time for nonnegative profits and in $O(n^6/\varepsilon)$ time for arbitrary integer profits. The second FPTAS runs instead in $O(n^5/\varepsilon^2)$ and $O(n^7/\varepsilon^2)$ time, respectively. The approximation guarantee, along with new structural insights, provides the bounds needed to control the scaled profit range and establish these running times.

Problem

• Colored Knapsack Problem
• approximation algorithm
• fully polynomial-time approximation schemes

Innovation

• approximation algorithm
• fully polynomial-time approximation schemes
• Colored Knapsack Problem