  We investigate the existence of approximation algorithms for maximization of
submodular functions, that run in fixed parameter tractable (FPT) time. Given a
non-decreasing submodular set function $v: 2^X \to \mathbb{R}$ the goal is to
select a subset $S$ of $K$ elements from $X$ such that $v(S)$ is maximized. We
identify three properties of set functions, referred to as $p$-separability
properties, and we argue that many real-life problems can be expressed as
maximization of submodular, $p$-separable functions, with low values of the
parameter $p$. We present FPT approximation schemes for the minimization and
maximization variants of the problem, for several parameters that depend on
characteristics of the optimized set function, such as $p$ and $K$. We confirm
that our algorithms are applicable to a broad class of problems, in particular
to problems from computational social choice, such as item selection or winner
determination under several multiwinner election systems.
