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.

翻译：我们研究预算化分层拟阵独立集问题。输入包括一个地面集，其中每个元素具有一个成本和一个非负利润，以及一个定义在这些元素上的分层拟阵和一个预算。目标是选择一个利润最大化的拟阵独立集，其总成本不超过预算。若干已知特例——例如无拟阵约束（经典背包问题）或均匀拟阵约束（带基数约束的背包问题）——均存在完全多项式时间近似方案（FPTAS）。相比之下，一般拟阵的预算化拟阵独立集（BMI）问题存在高效多项式时间近似方案（EPTAS），但不存在FPTAS。这意味着我们的问题存在EPTAS，这是本文之前已知的最优结果。我们提出预算化分层拟阵独立集的FPTAS，改进了针对此类拟阵族原有的EPTAS，并将已知的带基数约束背包问题和多选背包问题的FPTAS进行了推广。我们的方案基于一个利用分层拟阵树状结构的简单动态规划算法。