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Abstract

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].