  In the decremental Single-Source Shortest Path problem (SSSP), we are given a
weighted directed graph $G=(V,E,w)$ undergoing edge deletions and a source
vertex $r \in V$; let $n = |V|, m = |E|$ and $W$ be the aspect ratio of the
graph. The goal is to obtain a data structure that maintains shortest paths
from $r$ to all vertices in $V$ and can answer distance queries in $O(1)$ time,
as well as return the corresponding path $P$ in $O(|P|)$ time.
  This problem was first considered by Even and Shiloach [JACM'81], who
provided an algorithm with total update time $O(mn)$ for unweighted undirected
graphs; this was later extended to directed weighted graphs [FOCS'95, STOC'99].
There are conditional lower bounds showing that $O(mn)$ is in fact near-optimal
[ESA'04, FOCS'14, STOC'15, STOC'20]. In a breakthrough result, Forster et al.
showed that it is possible to achieve total update time $mn^{0.9+o(1)}\log W$
if the algorithm is allowed to return $(1+{\epsilon})$-approximate paths,
instead of exact ones [STOC'14, ICALP'15]. No further progress was made until
Probst Gutenberg and Wulff-Nilsen [SODA'20] provided a new approach for the
problem, which yields total time $\tilde{O}(\min{m^{2/3}n^{4/3}\log W,
(mn)^{7/8} \log W})$.
  Our result builds on this recent approach, but overcomes its limitations by
introducing a significantly more powerful abstraction, as well as a different
core subroutine. Our new framework yields a decremental
$(1+{\epsilon})$-approximate SSSP data structure with total update time
$\tilde{O}(n^2 \log^4 W)$. Our algorithm is thus near-optimal for dense graphs
with polynomial edge-weights. Our framework can also be applied to sparse
graphs to obtain total update time $\tilde{O}(mn^{2/3} \log^3 W)$.
  Our main technique allows us to convert SSSP algorithms for DAGs to ones for
general graphs, which we believe has significant potential to influence future
work.
