  We give the first almost-linear time algorithms for several problems in
incremental graphs including cycle detection, strongly connected component
maintenance, $s$-$t$ shortest path, maximum flow, and minimum-cost flow. To
solve these problems, we give a deterministic data structure that returns a
$m^{o(1)}$-approximate minimum-ratio cycle in fully dynamic graphs in amortized
$m^{o(1)}$ time per update. Combining this with the interior point method
framework of Brand-Liu-Sidford (STOC 2023) gives the first almost-linear time
algorithm for deciding the first update in an incremental graph after which the
cost of the minimum-cost flow attains value at most some given threshold $F$.
By rather direct reductions to minimum-cost flow, we are then able to solve the
problems in incremental graphs mentioned above.
  At a high level, our algorithm dynamizes the $\ell_1$ oblivious routing of
Rozho\v{n}-Grunau-Haeupler-Zuzic-Li (STOC 2022), and develops a method to
extract an approximate minimum ratio cycle from the structure of the oblivious
routing. To maintain the oblivious routing, we use tools from concurrent work
of Kyng-Meierhans-Probst Gutenberg which designed vertex sparsifiers for
shortest paths, in order to maintain a sparse neighborhood cover in fully
dynamic graphs.
  To find a cycle, we first show that an approximate minimum ratio cycle can be
represented as a fundamental cycle on a small set of trees resulting from the
oblivious routing. Then, we find a cycle whose quality is comparable to the
best tree cycle. This final cycle query step involves vertex and edge
sparsification procedures reminiscent of previous works, but crucially requires
a more powerful dynamic spanner which can handle far more edge insertions. We
build such a spanner via a construction that hearkens back to the classic
greedy spanner algorithm.
