AI 中文总结
研究有随机边长的有向图路径长度分布函数计算问题，证明其#P - 难，且在特定条件下对任意连续分布均成立，还表明该问题关于树宽k属于XP，给出i.i.d.均匀边长情况的动态规划算法及时复杂度。
AI 中文摘要
我们研究了在具有随机边长的有向图中计算最短和最长路径长度分布函数的问题。具体而言，当边长均匀分布时，该问题简化为计算由图结构定义的多面体的体积。我们证明，即使在随机边长根据具有某些自然条件（局部均匀性）的任何连续概率分布独立同分布的受限条件下，该问题也是#P - 难的。此硬度结果广泛适用：虽然均匀分布是简化的重要情况，但其他分布（如指数或正态分布）同样困难，因为它们在每个任意小的区间内都包含均匀分布。此外，我们表明该问题相对于基础无向图的树宽k属于XP。对于独立同分布均匀边长的特定情况，我们提出了一种新颖的动态规划算法，该算法通过迭代执行卷积来传播分布函数来处理树分解。我们的方法对于任何固定的树宽k实现了n^(O(k^2))的时间复杂度。
英文摘要
We investigate the problem of computing the distribution function for the shortest and longest path lengths in a directed graph with random edge lengths. Specifically, when these lengths are uniformly distributed, the problem reduces to computing the volume of a polytope defined by the graph structure. We establish that the problem is $\#P$-hard, even under the restricted condition that the random edge lengths are identically and independently distributed (i.i.d.) according to any continuous probability distribution with certain natural conditions, the local uniformity. This hardness result applies broadly: while the uniform distribution provides an essential case for the reduction, other distributions -- such as exponential or normal -- are similarly hard because they contain uniform distributions in every arbitrarily small interval. Furthermore, we show that the problem is contained within $\mathrm{XP}$ with respect to the treewidth $k$ of the underlying undirected graph. For the specific case of i.i.d. uniform edge lengths, we present a novel dynamic programming algorithm that processes a tree decomposition by iteratively performing convolutions to propagate distribution functions. Our approach achieves a time complexity of $n^{O(k^2)}$ for any fixed treewidth $k$.
