An EPTAS for Budgeted Matroid Independent Set

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 of the scheme is to find a representative set for the instance, whose cardinality depends solely on $1/\varepsilon$, where $\varepsilon > 0$ is the accuracy parameter of the scheme. The representative set is identified via matroid basis minimization, which can be solved by a simple greedy algorithm. Our scheme enumerates over subsets of the representative set and extends each subset using a linear program. The notion of representative sets may be useful in solving other variants of the budgeted matroid independent set problem.