Aaron Bernstein: Negative-Weight Single-Source Shortest Paths in Near-linear Time

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, can be negative, and are >= -W. This essentially resolves the classic negative-weight SSSP problem. The previous bounds are ~O((m+n^{1.5})\log W) and m^{4/3+o(1)}\log W. Near-linear time algorithms were known previously only for the special case of planar directed graphs. Also, independently of our work, the recent breakthrough on min-cost flow [Chen, Kyng, Liu,  Peng, Probst-Gutenberg and Sachdeva] implies an algorithm for negative-weight SSSP with running time m^{1+o(1)}.

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 O(m\sqrt{n}\log nW) bound from over three decades ago [Gabow and Tarjan SICOMP’89].