有向图中受限最短路径的确定性近线性时间近似算法
A Deterministic Near-Linear Time Approximation Algorithm for Restricted Shortest Paths in Directed Graphs
Vikrant Ashvinkumar
AI 速览：针对有向图受限最短路径问题，作者提出确定性双准则 (1+ε,1+ε) 近似算法，运行时间近线性，并可推广到多资源情形。
In the restricted shortest paths problem, we are given an $n$-vertex $m$-edge directed graph $G$ whose edges are assigned two non-negative weights (or resources) called lengths and costs, a source vertex $s$, and a cost threshold $C$. The goal is to find, for each target vertex $t$, the length of the shortest $(s,t)$-path whose total cost is at most $C$. Ashvinkumar, Bernstein, and Karczmarz (SODA 2025) gave the first randomized bicriteria $(1+\varepsilon,1+\varepsilon)$-approximate algorithm on directed graphs running in sub-$mn$ time, answering a problem posed by Bernstein (SODA 2012). We substantially improve this on two fronts, supplying a deterministic bicriteria $(1+\varepsilon,1+\varepsilon)$-approximate algorithm running in near-linear time. Our approach straightforwardly generalizes to more than two resources. For $k$ resources, a deterministic $(1+\varepsilon_1,\ldots,1+\varepsilon_k)$-approximate algorithm running in near-linear time is possible, with $\varepsilon_1=\ldots=\varepsilon_k=\varepsilon$.