用简单初等算法突破计数线性扩展的2^n障碍
Breaking the $2^n$ Barrier for Counting Linear Extensions with a Short Elementary Algorithm
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中文总结 AI 辅助
该研究提出确定性精确算法，以O^*(1.89^n)时间计数任意n元偏序集的线性扩展，突破2^n障碍并解决Koivisto的问题，改进Kozma的二维偏序集论证。
中文摘要 AI 辅助
有限偏序集的线性扩展是指符合该偏序关系的全序排列。我们提出一种确定性精确算法，可在O^*(1.89^n)时间内计数任意n元偏序集的线性扩展，其中O^*(\u00b7)表示忽略多项式因子。这突破了该通用问题的2^n障碍，解决了Koivisto在2013年达格施图尔研讨会明确提出的问题。该证明改进了Kozma针对二维偏序集的论证：当偏序集与反链差异足够大时，用链划分处理；否则，固定一个最大反链（即两两不可比元素的最大集合），对其中每个在反链外有可比上层元素的元素，仅记录首个出现的此类上层元素，通过解码引理从其重数枚举所得模式。模式确定后，每个反链元素有一个释放条件且至多有一个截止时间，动态规划仅存储各截止时间类中已释放元素的数量，用星号与横杠计数法界定总状态数。
英文摘要
A linear extension of a finite partially ordered set is a total ordering that respects the partial order. We give a deterministic exact algorithm that counts the linear extensions of an arbitrary $n$-element poset in time $O^*(1.89^n)$, where $O^*(\cdot)$ suppresses polynomial factors. This breaks the $2^n$ barrier for the general problem and resolves a question explicitly posed by Koivisto at Dagstuhl 2013. The proof refines an argument of Kozma for two-dimensional posets. A chain partition handles the case in which the poset is sufficiently far from an antichain. Otherwise, fix a maximum antichain (a largest set of pairwise incomparable elements). For each of its elements that has a comparable element above it outside the antichain, we record only which such element appears first. A decoding lemma enumerates the resulting patterns from their multiplicities. Once a pattern is fixed, each antichain element has a release condition and at most one deadline, so the dynamic program stores only the number of released elements in each deadline class. A stars-and-bars count bounds the total number of states.
