We study a family of matroid optimization problems with a linear constraint (MOL). In these problems, we seek a subset of elements which optimizes (i.e., maximizes or minimizes) a linear objective function subject to (i) a matroid independent set, or a matroid basis constraint, (ii) additional linear constraint. A notable member in this family is budgeted matroid independent set (BM), which can be viewed as classic $0/1$-knapsack with a matroid constraint. While special cases of BM, such as knapsack with cardinality constraint and multiple-choice knapsack, admit a fully polynomial-time approximation scheme (Fully PTAS), the best known result for BM on a general matroid is an Efficient PTAS. Prior to this work, the existence of a Fully PTAS for BM, and more generally, for any problem in the family of MOL problems, has been open. In this paper, we answer this question negatively by showing that none of the (non-trivial) problems in this family admits a Fully PTAS. This resolves the complexity status of several well studied problems. Our main result is obtained by showing first that exact weight matroid basis (EMB) does not admit a pseudo-polynomial time algorithm. This distinguishes EMB from the special cases of $k$-subset sum and EMB on a linear matroid, which are solvable in pseudo-polynomial time. We then obtain unconditional hardness results for the family of MOL problems in the oracle model (even if randomization is allowed), and show that the same results hold when the matroids are encoded as part of the input, assuming $P \neq NP$. For the hardness proof of EMB, we introduce the $\Pi$-matroid family. This intricate subclass of matroids, which exploits the interaction between a weight function and the matroid constraint, may find use in tackling other matroid optimization problems.

翻译：本文研究一类具有线性约束的拟阵优化问题（MOL）。在这类问题中，我们寻求一个子集元素，该子集在满足（i）拟阵独立集或拟阵基约束和（ii）附加线性约束的条件下，优化（即最大化或最小化）线性目标函数。该问题族中一个值得关注的成员是预算约束拟阵独立集（BM），可视为经典$0/1$-背包问题附加拟阵约束。虽然BM的特殊情形（如基数约束背包问题和多选择背包问题）允许完全多项式时间近似方案（Fully PTAS），但一般拟阵上BM的最佳已知结果仅为高效多项式时间近似方案（Efficient PTAS）。在此研究之前，BM是否存在Fully PTAS（更广泛地，MOL问题族中任意问题是否存在Fully PTAS）一直未解。本文通过证明该问题族中所有（非平凡）问题均不允许Fully PTAS，否定了这一可能性。这一结果解决了若干经典问题的复杂性状态。我们首先证明精确权重拟阵基（EMB）不存在伪多项式时间算法，从而获得了主要结论。这使EMB区别于可伪多项式时间求解的$k$-子集和问题及线性拟阵上的EMB等特殊情形。随后我们在预言机模型下（即使允许随机化）获得MOL问题族的无条件困难性结果，并证明当拟阵作为输入一部分编码时（假设$P \neq NP$），相同结论依然成立。在EMB的困难性证明中，我们引入$\Pi$-拟阵族。这一精心构造的拟阵子类通过利用权重函数与拟阵约束之间的交互作用，可能为其他拟阵优化问题的求解提供新途径。