  All roads lead to Rome is the core idea of the puzzle game Roma. It is played
on an $n \times n$ grid consisting of quadratic cells. Those cells are grouped
into boxes of at most four neighboring cells and are either filled, or to be
filled, with arrows pointing in cardinal directions. The goal of the game is to
fill the empty cells with arrows such that each box contains at most one arrow
of each direction and regardless where we start, if we follow the arrows in the
cells, we will always end up in the special Roma-cell. In this work, we study
the computational complexity of the puzzle game Roma and show that completing a
Roma board according to the rules is an \NP-complete task, counting the number
of valid completions is #Ptime-complete, and determining the number of preset
arrows needed to make the instance \emph{uniquely} solvable is
$\Sigma_2^P$-complete. We further show that the problem of completing a given
Roma instance on an $n\times n$ board cannot be solved in time
$\mathcal{O}\left(2^{{o}(n)}\right)$ under ETH and give a matching dynamic
programming algorithm based on the idea of Catalan structures.
