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 and can be negative. 1 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].