  We study budgeted variants of well known maximization problems with multiple
matroid constraints. Given an $\ell$-matchoid $\cm$ on a ground set $E$, a
profit function $p:E \rightarrow \mathbb{R}_{\geq 0}$, a cost function $c:E
\rightarrow \mathbb{R}_{\geq 0}$, and a budget $B \in \mathbb{R}_{\geq 0}$, the
goal is to find in the $\ell$-matchoid a feasible set $S$ of maximum profit
$p(S)$ subject to the budget constraint, i.e., $c(S) \leq B$. The {\em budgeted
$\ell$-matchoid} (BM) problem includes as special cases budgeted
$\ell$-dimensional matching and budgeted $\ell$-matroid intersection. A strong
motivation for studying BM from parameterized viewpoint comes from the
APX-hardness of unbudgeted $\ell$-dimensional matching (i.e., $B = \infty$)
already for $\ell = 3$. Nevertheless, while there are known FPT algorithms for
the unbudgeted variants of the above problems, the {\em budgeted} variants are
studied here for the first time through the lens of parameterized complexity.
  We show that BM parametrized by solution size is $W[1]$-hard, already with a
degenerate single matroid constraint. Thus, an exact parameterized algorithm is
unlikely to exist, motivating the study of {\em FPT-approximation schemes}
(FPAS). Our main result is an FPAS for BM (implying an FPAS for
$\ell$-dimensional matching and budgeted $\ell$-matroid intersection), relying
on the notion of representative set $-$ a small cardinality subset of elements
which preserves the optimum up to a small factor. We also give a lower bound on
the minimum possible size of a representative set which can be computed in
polynomial time.
