  We study the budgeted versions of the well known matching and matroid
intersection problems. While both problems admit a polynomial-time
approximation scheme (PTAS) [Berger et al. (Math. Programming, 2011), Chekuri,
Vondrak and Zenklusen (SODA 2011)], it has been an intriguing open question
whether these problems admit a fully PTAS (FPTAS), or even an efficient PTAS
(EPTAS).
  In this paper we answer the second part of this question affirmatively, by
presenting an EPTAS for budgeted matching and budgeted matroid intersection. A
main component of our scheme is a novel construction of representative sets for
desired solutions, whose cardinality depends only on $\varepsilon$, the
accuracy parameter. Thus, enumerating over solutions within a representative
set leads to an EPTAS. This crucially distinguishes our algorithms from
previous approaches, which rely on exhaustive enumeration over the solution
set. Our ideas for constructing representative sets may find use in tackling
other budgeted optimization problems, and are thus of independent interest.
