  Gradual semantics within abstract argumentation associate a numeric score
with every argument in a system, which represents the level of acceptability of
this argument, and from which a preference ordering over arguments can be
derived. While some semantics operate over standard argumentation frameworks,
many utilise a weighted framework, where a numeric initial weight is associated
with each argument. Recent work has examined the inverse problem within gradual
semantics. Rather than determining a preference ordering given an argumentation
framework and a semantics, the inverse problem takes an argumentation
framework, a gradual semantics, and a preference ordering as inputs, and
identifies what weights are needed to over arguments in the framework to obtain
the desired preference ordering. Existing work has attacked the inverse problem
numerically, using a root finding algorithm (the bisection method) to identify
appropriate initial weights. In this paper we demonstrate that for a class of
gradual semantics, an analytical approach can be used to solve the inverse
problem. Unlike the current state-of-the-art, such an analytic approach can
rapidly find a solution, and is guaranteed to do so. In obtaining this result,
we are able to prove several important properties which previous work had posed
as conjectures.
