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.

翻译：我们研究了著名的匹配和拟阵交问题的预算版本。尽管这两个问题均承认多项式时间近似方案（PTAS）[Berger 等人 (Math. Programming, 2011)，Chekuri、Vondrak 和 Zenklusen (SODA 2011)]，但它们是否承认完全多项式时间近似方案（FPTAS），甚至有效多项式时间近似方案（EPTAS），一直是一个引人入胜的开放问题。在本文中，我们通过为预算匹配和预算拟阵交提出一种EPTAS，肯定地回答了该问题的第二部分。我们方案的一个主要组成部分是针对期望解集的一种新颖的代表集构造方法，其基数仅依赖于精度参数 $\varepsilon$。因此，在代表集内枚举解即可导出EPTAS。这关键性地使我们的算法区别于先前依赖对解集进行穷举枚举的方法。我们构造代表集的思路可能有助于解决其他预算优化问题，因此具有独立的研究价值。