线性扩张的直径对
Diametral Pairs of Linear Extensions
摘要：
给定一个有限偏序集P，我们考虑P的具有最大距离的线性扩张对，其中两个线性扩张L_1，L_2之间的距离是P的元素对在L_1，L_2中以不同顺序出现的个数.直径对使P的所有线性扩张对之间的距离最大化。我们表明，计算线性扩张直径一般是NP-完全的，但可以解决在多项式时间的偏序集的宽度为3。费尔斯纳和费尔特证明，在每一个直径对中，至少有一个线性扩张反转了一个临界对。我们构造一个反例来证明这个猜想。另一方面，我们发现一个稍微强一些的性质适用于许多类的偏序集：我们称一个偏序集为“直径反转”，如果在每一个直径对，两个线性扩展都反转一个临界对。在其他结果中，我们证明了区间序和3层偏序集是直径反转的。从后者可以得出几乎所有偏序集都是直径反转的。
英文摘要：
Given a finite poset P, we consider pairs of linear extensions of P with maximal distance, where the distance between two linear extensions L_1, L_2 is the number of pairs of elements of P appearing in different orders in L_1 and L_2. A diametral pair maximizes the distance among all pairs of linear extensions of P. Felsner and Reuter defined the linear extension diameter of P as the distance between a diametral pair of linear extensions. We show that computing the linear extension diameter is NP-complete in general, but can be solved in polynomial time for posets of width 3. Felsner and Reuter conjectured that, in every diametral pair, at least one of the linear extensions reverses a critical pair. We construct a counterexample to this conjecture. On the other hand, we show that a slightly stronger property holds for many classes of posets: We call a poset "diametrally reversing" if, in every diametral pair, both linear extensions reverse a critical pair. Among other results we show that interval orders and 3-layer posets are diametrally reversing. From the latter it follows that almost all posets are diametrally reversing.
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