  A recent breakthrough by K\"unnemann, Mazowiecki, Sch\"utze, Sinclair-Banks,
and Wegrzycki (ICALP, 2023) bounds the running time for the coverability
problem in $d$-dimensional vector addition systems under unary encoding to
$n^{2^{O(d)}}$, improving on Rackoff's $n^{2^{O(d\lg d)}}$ upper bound (Theor.
Comput. Sci., 1978), and provides conditional matching lower bounds.
  In this paper, we revisit Lazi\'c and Schmitz' "ideal view" of the backward
coverability algorithm (Inform. Comput., 2021) in the light of this
breakthrough. We show that the controlled strongly monotone descending chains
of downwards-closed sets over $\mathbb{N}^d$ that arise from the dual backward
coverability algorithm of Lazi\'c and Schmitz on $d$-dimensional unary vector
addition systems also enjoy this tight $n^{2^{O(d)}}$ upper bound on their
length, and that this also translates into the same bound on the running time
of the backward coverability algorithm.
  Furthermore, our analysis takes place in a more general setting than that of
Lazi\'c and Schmitz, which allows to show the same results and improve on the
2EXPSPACE upper bound derived by Benedikt, Duff, Sharad, and Worrell (LICS,
2017) for the coverability problem in invertible affine nets.

Data Structures and Algorithms;Computational Complexity;Logic in Computer Science