AI 中文总结
该研究针对有向图的分数多数着色问题，通过随机循环匹配采样等方法改进了着色数上界，证明了相关判定问题的NP完全性及不可近似性阈值，还得到了随机近似算法。
AI 中文摘要
有向图中的一组顶点是多数稳定的，当且仅当该组中每个顶点的出邻接点至多有一半在该组内。分数多数着色为这类集合分配非负权重，要求每个顶点被覆盖的总权重至少为1；其最小总权重即为分数多数着色数。我们证明，每个有限无环有向图的分数多数着色数至多为523/140<3.736，改进了Anastos、Lamaison、Steiner和Szabó给出的3.9602的界。我们的构造方法是：从随机循环匹配的对中进行排他采样，随后对无环剩余部分进行删除和修正。相同方法可得到以下界：含定向环因子的有向图的分数多数着色数低于3.430，含偶环因子的有向图低于3.287，锦标赛图低于3.324。由此，对任意固定的ε>0，我们得到一个随机(2.491+ε)-近似算法，可在期望多项式时间内生成显式的分数多数着色。在复杂性方面，我们证明判定分数多数着色数是否等于3/2是NP完全问题，并确定了乘法不可近似性阈值72/71和加性上估计阈值3/142。所有三个结果均适用于出度为0或2、且每条有向路径长度至多为2的无环定向有向图。
英文摘要
A set of vertices in a digraph is majority-stable if each of its vertices has at most half of its outneighbors in the set. A fractional majority coloring assigns nonnegative weights to such sets, covering each vertex to total weight at least one; its minimum total weight is the fractional majority coloring number. We prove that every finite loopless digraph has fractional majority coloring number at most \(523/140<3.736\), improving the bound \(3.9602\) of Anastos, Lamaison, Steiner and Szabó. Our construction uses exclusive sampling from pairs in random cycle matchings, followed by deletion and a correction on the acyclic remainder. The same approach yields bounds below \(3.430\) for digraphs with a directed cycle factor, \(3.287\) for those with an even cycle factor, and \(3.324\) for tournaments. As a consequence, for every fixed \(\varepsilon>0\), we obtain a randomized \((2.491+\varepsilon)\)-approximation algorithm producing an explicit fractional majority coloring in expected polynomial time. On the complexity side, we prove NP-completeness of deciding whether the fractional majority coloring number equals $3/2$, and establish a multiplicative inapproximability threshold of $72/71$ and an additive upper-estimation threshold of $3/142$. All three results hold for acyclic oriented digraphs with outdegrees zero or two in which every directed path has length at most two.
