彩色背包问题的紧1/3近似算法与完全多项式时间近似方案

中文摘要 AI 辅助

彩色背包问题（Colored Knapsack Problem, ColKP）推广了经典背包问题，它将物品划分为颜色类别，并要求所选物品存在一个排序，使得连续物品具有不同颜色。该问题是弱NP-hard的，文献中提出了两种伪多项式动态规划（DP）算法。这两种DP算法的最坏情况运行时间分别为O(b n^4)和O(b^2 n^3)，其中b是背包容量，n是物品数量。我们开发了ColKP的第一个近似算法。通过对其自然整数规划形式的线性规划松弛的最优基本解进行舍入，并修复颜色可行性，我们获得了一个线性时间近似算法，其最坏情况性能比为1/3。然后，我们重新表述了两种DP算法，使得利润（而非背包容量）作为其伪多项式维度的索引，并将其与利润缩放相结合，得到了两个完全多项式时间近似方案（FPTAS）。第一个FPTAS在非负利润下运行时间为O(n^5/ε)，在任意整数利润下运行时间为O(n^6/ε)。第二个FPTAS的运行时间分别为O(n^5/ε^2)和O(n^7/ε^2)。近似保证以及新的结构洞见提供了控制缩放利润范围并确立这些运行时间所需的界限。

英文摘要

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.

发表机构

• Sapienza University of Rome（罗马第一大学）

机构由 AI 辅助整理，请以论文原文为准。