优先约束背包问题中的最优宏项序列
Parametric maximum closure on precedence forests
AI总结：
研究优先约束背包问题线性规划松弛的最优解结构，引入宏项和可行序列概念，证明最优解由宏项序列完全刻画，并针对森林结构提出O(n^2)算法。
摘要：
优先约束背包问题（PCKP）要求在背包容量约束和由有向无环图编码的优先约束下，寻找最大利润的物品子集。我们研究了PCKP自然整数线性规划公式的线性规划（LP）松弛的最优解结构。我们引入了宏项和宏项的可行序列的概念，该序列在尊重优先结构的同时划分物品集。我们证明了最优LP解由宏项的最优序列完全刻画：物品按其宏项的利润-重量比非递增顺序打包，最多有一个宏项被部分包含。我们进一步证明了容量约束的参数拉格朗日函数的断点与最优序列中宏项的利润-重量比一致，并提供了每个宏项内可行流形式的最优对偶解的完整组合刻画。最后，对于优先图是森林的特殊情况，我们设计了一个O(n^2)算法来计算最优序列，对于内树或外树，该算法改进为O(n log n)，其中n表示物品数量。
英文摘要：
The Maximum Closure Problem asks for a maximum-weight closed subset of a precedence-constrained set of vertices; when vertex weights depend affinely on a scalar parameter $λ$, as in open-pit mine scheduling, the goal becomes computing the optimal closed set for every value of $λ$ at once. We address this problem when the precedence graph is a directed forest. We present two algorithms (Peel-and-Contract, PaC, and its dual DPaC) that compute the full parametric solution -- a canonical sequence of disjoint closure layers inducing the nested optimal closures -- by iteratively aggregating vertices along precedence arcs, with time-complexity $\mathcal{O}(n^2)$ for general forests and space-complexity $\mathcal{O}(n)$; heap-based implementations (HPaC, DHPaC) have a worse time-complexity of $\mathcal{O}(n^2\log n)$, but on random instances their computing times empirically grow as $n\log n$. When the forest is restricted to in-trees or out-trees, specialized heap-based variants (HIPaC, HOPaC) achieve time-complexity $\mathcal{O}(n\log n)$. We then present a further aggregation-based algorithm (Rake-and-Compress, RaC) with time- and space-complexity $\mathcal{O}(n\log n)$ also for general forests, which matches the $Ω(n\log n)$ lower bound for comparison-based algorithms. Computational experiments on random and structured forests with up to $10^5$ vertices characterize the practical behavior of the proposed algorithms, identify the topologies on which HPaC and RaC are preferable, and compare HPaC with the public implementation of the fully parametric pseudoflow algorithm, the state of the art for arbitrary precedence graphs, which it outperforms on forests by one to more than three orders of magnitude.
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