🤖 AI Summary
This paper addresses the single-source shortest paths (SSSP) problem with negative-weight edges on general directed graphs, achieving the first near-linear-time algorithm. For three decades, the best-known combinatorial lower bound stood at $O(msqrt{n}log W)$; this work breaks that barrier with the first purely combinatorial, non-optimization-based, non-dynamic randomized algorithm. Leveraging novel graph decomposition techniques and specialized handling of integer edge weights, it attains a running time of $O(m log^8 n log W)$. This result substantially improves upon the two best prior algorithms presented at FOCS’20 and resolves a long-standing complexity bottleneck for negative-weight SSSP. Crucially, it is the first combinatorial algorithm to break the $ ilde{O}(msqrt{n})$ barrier on general graphs, marking a fundamental advance in the theoretical understanding of shortest-path computation under negative weights.
📝 Abstract
We present a randomized algorithm that computes single-source shortest paths (SSSP) in $Oleft(m log ^{8}(n) log W
ight)$ time when edge weights are integral and can be negative. 1 This essentially resolves the classic negative-weight SSSP problem. The previous bounds are $ ilde{O}left(left(m+n^{1.5}
ight) log W
ight)$ [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 $O(m sqrt{n} log W)$ bound from over three decades ago [Gabow and Tarjan SICOMP’89].
