An EPTAS for Budgeted Matroid Independent Set
Ilan Doron-Arad, Ariel Kulik, Hadas Shachnai

TL;DR
This paper presents an Efficient Polynomial-Time Approximation Scheme (EPTAS) for the budgeted matroid independent set problem, improving upon prior PTAS results by using representative sets and matroid basis minimization.
Contribution
The authors develop the first EPTAS for the problem, utilizing a novel approach with representative sets and greedy algorithms for matroid basis minimization.
Findings
Achieved an EPTAS for the budgeted matroid independent set problem.
The scheme relies on a representative set with size depending only on 1/ε.
The approach extends to other variants of the problem.
Abstract
We consider the budgeted matroid independent set problem. The input is a ground set, where each element has a cost and a non-negative profit, along with a matroid over the elements and a budget. The goal is to select a subset of elements which maximizes the total profit subject to the matroid and budget constraints. Several well known special cases, where we have, e.g., a uniform matroid and a budget, or no matroid constraint (i.e., the classic knapsack problem), admit a fully polynomial-time approximation scheme (FPTAS). In contrast, already a slight generalization to the multi-budgeted matroid independent set problem has a PTAS but does not admit an efficient polynomial-time approximation scheme (EPTAS). This implies a PTAS for our problem, which is the best known result prior to this work. Our main contribution is an EPTAS for the budgeted matroid independent set problem. A key idea…
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