In this paper we derive tight lower bounds resolving the hardness status of several fundamental weighted matroid problems. One notable example is budgeted matroid independent set, for which we show there is no fully polynomial-time approximation scheme (FPTAS), indicating the Efficient PTAS of [Doron-Arad, Kulik and Shachnai, SOSA 2023] is the best possible. Furthermore, we show that there is no pseudo-polynomial time algorithm for exact weight matroid independent set, implying the algorithm of [Camerini, Galbiati and Maffioli, J. Algorithms 1992] for representable matroids cannot be generalized to arbitrary matroids. Similarly, we show there is no Fully PTAS for constrained minimum basis of a matroid and knapsack cover with a matroid, implying the existing Efficient PTAS for the former is optimal. For all of the above problems, we obtain unconditional lower bounds in the oracle model, where the independent sets of the matroid can be accessed only via a membership oracle. We complement these results by showing that the same lower bounds hold under standard complexity assumptions, even if the matroid is encoded as part of the instance. All of our bounds are based on a specifically structured family of paving matroids.

翻译：本文推导了若干基础加权拟阵问题的紧下界，明确了其计算复杂度的硬度状态。一个典型例子是预算拟阵独立集问题：我们证明其不存在完全多项式时间近似方案（FPTAS），表明[Doron-Arad, Kulik 和 Shachnai, SOSA 2023] 中的高效PTAS已是最优结果。进一步地，我们证明精确权重拟阵独立集问题不存在伪多项式时间算法，这意味着[Camerini, Galbiati 和 Maffioli, J. Algorithms 1992] 中针对可表示拟阵的算法无法推广至任意拟阵。类似地，我们证明约束拟阵最小基与带拟阵的背包覆盖问题均不存在完全FPTAS，表明现有关于前者的高效PTAS已达到最优。针对上述所有问题，我们在预言机模型中获得了无条件下界——其中拟阵的独立集仅能通过成员资格预言机访问。作为补充，我们证明即使拟阵作为实例编码存在，在标准复杂度假设下上述下界依然成立。所有下界均基于一类具有特殊结构的铺砌拟阵。