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We study the budgeted versions of the well known matching and matroidintersection problems. While both problems admit a polynomial-timeapproximation scheme (PTAS) [Berger et al. (Math. Programming, 2011), Chekuri,Vondrak and Zenklusen (SODA 2011)], it has been an intriguing open questionwhether 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, bypresenting an EPTAS for budgeted matching and budgeted matroid intersection. Amain component of our scheme is a novel construction of representative sets fordesired solutions, whose cardinality depends only on $\varepsilon$, theaccuracy parameter. Thus, enumerating over solutions within a representativeset leads to an EPTAS. This crucially distinguishes our algorithms fromprevious approaches, which rely on exhaustive enumeration over the solutionset. Our ideas for constructing representative sets may find use in tacklingother budgeted optimization problems, and are thus of independent interest.

An EPTAS for Budgeted Matching and Budgeted Matroid Intersection