An improved FPTAS for budgeted laminar matroid independent set

Abstract

The budgeted laminar matroid independent set problem seeks to maximize the total profit of a cardinality constrained independent set within a laminar matroid, subject to a budget on total cost. A pseudo-polynomial DP algorithm and an FPTAS were recently proposed for this problem. We extract properties of these algorithms and reduce the running time of the FPTAS from $O\left({\left|S\right|}^7{\epsilon }^{-2}\right)$ to $O({\left|S\right|}^5{\epsilon }^{-2}+{\left|S\right|}^5{\log }_2\left|S\right|)$, where S is the ground set and ε is the relative error.