AI 中文总结
研究边权为特定线性形式的有向图的参数最短路径问题，给出\(n^{O(d\log n)}\)上界指数级改进前人结果，还适用于正边权无向图及特定有向图，构建了最短路径预言机。
AI 中文摘要
我们研究边权形式为\(\mathsf{wt}(e) = a_{e,1} \lambda_1 + a_{e,2} \lambda_2 + a_{e,3} \lambda_3 + \cdots + a_{e,d} \lambda_d + a_{e,d + 1}\)的有向图中的最短路径。这被称为参数最短路径问题，自20世纪80年代起就被研究。对于\(d = 1\)，Carstensen给出了\(n\)顶点图中最短路径数量的上下界；对于\(d = 2\)，Gajjar和Radhakrishnan给出了上界，Barth等人将其推广到所有正整数\(d\)。多年来下界未改进。本文给出所有正整数\(d\)的\(n^{O(d\log n)}\)上界，指数级改进了之前上界，还证明可用于正边权无向图，对于边权为至多\(q\)次单变量多项式的有向图给出\(n^{O(\log{n}+\log{q})}\)上界，最后构建了一个最短路径预言机。
英文摘要
We study shortest paths in directed graphs whose edge weights are of the form $$ \mathsf{wt}(e) = a_{e,1} λ_1 + a_{e,2} λ_2 + a_{e,3} λ_3 + \cdots + a_{e,d} λ_d + a_{e,d+1}.$$ Here, each $a_{e,i}\in\mathbb{R}$ is a fixed constant for each edge $e$, whereas each $λ_i\in\mathbb{R}$ is common across the entire graph. So, there could be different shortest paths in the graph for different values of the $λ_i$'s. This is called the Parametric Shortest Paths problem, and has been studied since the 1980s. For $d=1$, Carstensen (1983) showed that the number of shortest paths in $n$-vertex graphs is at most $n^{O(\log n)}$. She also proved a matching lower bound of $n^{Ω(\log n)}$, later refined by Mulmuley & Shah (2001). For $d=2$, Gajjar & Radhakrishnan (2019) showed an upper bound of $n^{O(\log^2 n)}$. Barth, Funke & Proissl (2022) generalized their result to prove an upper bound of $n^{O_d(\log^d n)}$ for all positive integers $d$. The lower bound did not undergo any improvement over the years. In this paper, we close this long line of research by showing an $n^{O(d\log n)}$ upper bound for all positive integers $d$, exponentially improving the previous upper bound. We observe that a matching lower bound of $n^{Ω(d\log n)}$ can be obtained from earlier works. We also show that our proof can be adapted to work for undirected graphs with positive edge weights. Furthermore, for directed graphs whose edge weights are univariate polynomials of degree at most $q$, we prove an upper bound of $n^{O(\log{n}+\log{q})}$. Finally, building upon work on the Point Location problem by Ezra, Har-Peled, Kaplan & Sharir (2020), we construct a Shortest Path Oracle which takes as input a point $\overline{x}\in \mathbb{R}^d$, and outputs a shortest path at $\overlineλ=\overline{x}$ in sublinear time (for a wide regime of $d$).
Comments21 pages, 6 figures; to be presented at FOCS 2026
