  The signed double Roman domination problem is a combinatorial optimization
problem on a graph asking to assign a label from $\{\pm{}1,2,3\}$ to each
vertex feasibly, such that the total sum of assigned labels is minimized. Here
feasibility is given whenever (i) vertices labeled $\pm{}1$ have at least one
neighbor with label in $\{2,3\}$; (ii) each vertex labeled $-1$ has one
$3$-labeled neighbor or at least two $2$-labeled neighbors; and (iii) the sum
of labels over the closed neighborhood of any vertex is positive. The
cumulative weight of an optimal labeling is called signed double Roman
domination number (SDRDN). In this work, we first consider the problem on
general cubic graphs of order $n$ for which we present a sharp $n/2+\Theta(1)$
lower bound for the SDRDN by means of the discharging method. Moreover, we
derive a new best upper bound. Observing that we are often able to minimize the
SDRDN over the class of cubic graphs of a fixed order, we then study in this
context generalized Petersen graphs for independent interest, for which we
propose a constraint programming guided proof. We then use these insights to
determine the SDRDNs of subcubic $2\times m$ grid graphs, among other results.
