两个容量下的逆背包问题：哪些价值-基数凸包对是可实现的？
Inverse knapsack at two capacities: which pairs of value-cardinality hulls are realizable?
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中文总结 AI 辅助
本文研究两个容量下逆背包问题中价值-基数凸包对的实现性，提出交换闭包条件，并证明线性闭包严格更大，给出四带分类和复杂度结果。
中文摘要 AI 辅助
一个物品集在两个容量 $R<D$ 下评估，产生两个关于基数的最优价值凹包。我们询问哪些指定的对会出现。交换论证给出了顶点见证上的一个必要条件系统，即交换闭包（EC），其标量推论形成线性闭包。我们展示了一个满足所有标量检验但失败 EC 的对，因此线性闭包在更大的终端计数三时已经严格更大；以及一个全局一致的 EC 见证，尽管其目标对是可实现的，却不存在共同大小的表示。在基数上限下，一个家族的四带分类给出了上限四可实现性、无上限可实现性和仅顶点的有上限闭包的精确阈值。较大终端计数至多为二的对已被刻画。在显式编码下，决策问题属于 $Σ_2^p$，并且对于固定的终端基数在多项式时间内可解。精确的有限证书确立了在六个指定域上标量和见证条件的一致性；无上限 EC 的充分性仍然开放。
英文摘要
One item set evaluated at two capacities $R<D$ produces two concave hulls of optimal value against cardinality. We ask which prescribed pairs arise. Exchange arguments give a necessary system on vertex witnesses, exchange closure (EC), whose scalar consequences form the linear closure. We exhibit a pair satisfying every scalar test that fails EC, so the linear closure is strictly larger already at larger terminal count three; and a globally coherent EC witness admitting no common-size representation although its target pair is realizable. Under a cardinality cap, a four-band classification of one family gives exact thresholds for cap-four realizability, uncapped realizability and the vertex-only capped closure. Pairs whose larger terminal count is at most two are characterized. Under the explicit encoding the decision problem lies in $Σ_2^p$ and is polynomial-time for fixed terminal cardinalities. Exact finite certificates establish agreement of the scalar and witness conditions on six specified domains; sufficiency of uncapped EC remains open.
发表机构
- University of Notre Dame（圣母大学）
机构由 AI 辅助整理，请以论文原文为准。
