  We analyze the sketching approximability of constraint satisfaction problems
on Boolean domains, where the constraints are balanced linear threshold
functions applied to literals. In~particular, we explore the approximability of
monarchy-like functions where the value of the function is determined by a
weighted combination of the vote of the first variable (the president) and the
sum of the votes of all remaining variables. The pure version of this function
is when the president can only be overruled by when all remaining variables
agree. For every $k \geq 5$, we show that CSPs where the underlying predicate
is a pure monarchy function on $k$ variables have no non-trivial sketching
approximation algorithm in $o(\sqrt{n})$ space. We also show infinitely many
weaker monarchy functions for which CSPs using such constraints are
non-trivially approximable by $O(\log(n))$ space sketching algorithms.
Moreover, we give the first example of sketching approximable asymmetric
Boolean CSPs. Our results work within the framework of Chou, Golovnev, Sudan,
and Velusamy (FOCS 2021) that characterizes the sketching approximability of
all CSPs. Their framework can be applied naturally to get a computer-aided
analysis of the approximability of any specific constraint satisfaction
problem. The novelty of our work is in using their work to get an analysis that
applies to infinitely many problems simultaneously.
