An FPTAS for Budgeted Laminar Matroid Independent Set

Abstract

We study the budgeted laminar matroid independent set problem. The input is a ground set, where each element has a cost and a non-negative profit, along with a laminar matroid over the elements and a budget. The goal is to select a maximum profit independent set of the matroid whose total cost is bounded by the budget. Several well known special cases, where we have, e.g., no matroid constraint (the classic knapsack problem) or a uniform matroid constraint (knapsack with a cardinality constraint), admit a fully polynomial-time approximation scheme (FPTAS). In contrast, the budgeted matroid independent set (BMI) problem with a general matroid has an efficient polynomial-time approximation scheme (EPTAS) but does not admit an FPTAS. This implies an EPTAS for our problem, which is the best known result prior to this work. We present an FPTAS for budgeted laminar matroid independent set, improving the previous EPTAS for this matroid family and generalizing the FPTAS known for knapsack with a cardinality constraint and multiple-choice knapsack. Our scheme is based on a simple dynamic program which utilizes the tree-like structure of laminar matroids.