Negative-Weight Single-Source Shortest Paths in Almost-linear Time

We present a randomized algorithm that computes single-source shortest paths (SSSP) in $m^{1+o(1)}\log W$ time when edge weights are integral and can be negative; here, $n$ and $m$ denote the number of vertices and edges, respectively, and $W\geq 2$ is such that every edge weight is at least $-W$; $\tilde O$ hides polylogarithmic factors. This essentially resolves the classic negative-weight SSSP problem. The previous bounds are $\tilde O((m^{1}+n^{1.5})\log W)$ [BLNPSSSW FOCS'20] and $m^{4/3+o(1)}\log W$ [AMV FOCS'20]. In contrast to all recent developments that rely on sophisticated continuous optimization methods and dynamic algorithms, our algorithm is based on 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]. Beside being combinatorial, an important feature of our algorithm is in its simplicity: treating our graph decomposition as a black-box, we believe that the reader can reconstruct our algorithm and analysis from our 6-page overview. Independently from our result, the recent major breakthrough by Chen, Kyng, Liu, Peng, Gutenberg, and Sachdeva [ChenKLPGS22] achieve an almost-linear time bound for min-cost flow, implying the same bound for our problem. We discuss this result at the end of the introduction.