We study the problem of estimating a vertex's PageRank within a constant relative error, with constant probability. We prove that an adaptive variant of the simple classic bidirectional algorithm is instance-optimal up to a polylogarithmic factor for all directed graphs of order $n$ whose maximum in- and out-degrees are at most a constant fraction of $n$. In other words, there is no correct algorithm that can be faster than our algorithm on any such graph by more than a polylogarithmic factor. We further extend the instance-optimality to all graphs in which at most a polylogarithmic number of vertices have unbounded degrees. This covers all sparse graphs with $\tilde{O}(n)$ edges. In addition, we provide a counterexample showing that the bidirectional algorithm is not instance-optimal for graphs whose degrees are mostly equal to $n$. We also consider weighted graphs and multigraphs. We show that the bidirectional algorithm is instance-optimal on \emph{all} multigraphs, but for weighted simple graphs, we have almost the same limitations as for unweighted simple graphs.

翻译：我们研究在恒定相对误差和恒定概率下估计顶点PageRank值的问题。我们证明，经典简单双向算法的一种自适应变体在所有最大入度和出度至多为$n$的常数倍的有向图（阶数为$n$）上，在多项式对数因子内达到实例最优。换言之，对于任何此类图，不存在正确算法能比我们的算法快超过一个多项式对数因子。我们进一步将实例最优性推广到至多有对数多项式个顶点具有无界度的所有图。这涵盖了所有具有$\tilde{O}(n)$条边的稀疏图。此外，我们给出一个反例，表明双向算法在大多数度数等于$n$的图上并非实例最优。我们还考虑了加权图和多图。我们证明，双向算法在\emph{所有}多图上都是实例最优的，但对于加权简单图，我们面临与未加权简单图几乎相同的限制。