We present a randomized algorithm that computes single-source shortest paths (SSSP) in $O(m\log^8(n)\log W)$ time when edge weights are integral and can be negative. This essentially resolves the classic negative-weight SSSP problem. The previous bounds are $\tilde O((m+n^{1.5})\log W)$ [BLNPSSSW FOCS'20] and $m^{4/3+o(1)}\log W$ [AMV FOCS'20]. Near-linear time algorithms were known previously only for the special case of planar directed graphs [Fakcharoenphol and Rao FOCS'01]. In contrast to all recent developments that rely on sophisticated continuous optimization methods and dynamic algorithms, our algorithm is simple: it requires only a simple graph decomposition and elementary combinatorial tools. In fact, ours is the first combinatorial algorithm for negative-weight SSSP to break through the classic $\tilde O(m\sqrt{n}\log W)$ bound from over three decades ago [Gabow and Tarjan SICOMP'89].

翻译：我们提出了一种随机算法，可在 $O(m\log^8(n)\log W)$ 时间内计算边权为整数且可负的单源最短路径（SSSP）。这本质上解决了经典的负权SSSP问题。此前的最优边界为 $\tilde O((m+n^{1.5})\log W)$ [BLNPSSSW FOCS'20] 和 $m^{4/3+o(1)}\log W$ [AMV FOCS'20]。近线性时间算法此前仅针对平面有向图的特例 [Fakcharoenphol and Rao FOCS'01]。与所有依赖复杂连续优化方法和动态算法的近期进展不同，我们的算法简单易懂：仅需简单的图分解和基础组合工具。事实上，这是首个突破三十多年前经典 $\tilde O(m\sqrt{n}\log W)$ 边界 [Gabow and Tarjan SICOMP'89] 的负权SSSP组合算法。