发表机构
MIT; Technion（麻省理工学院; 以色列理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究带线性约束的拟阵优化问题（P-MOL），提出统一EPTAS算法，以(1±ε)近似比解决所有非平凡变体，并证明拟阵交覆盖变体不存在EPTAS，揭示覆盖与预算约束的定性差异。
AI 中文摘要
我们研究以下带线性约束的拟阵优化问题类别（P-MOL）。给定一个拟阵 M=(E,I)、两个权重函数 v,w:E→R≥0 以及一个阈值 L∈R≥0，求 opt v(S)，其中 S 是 M 的独立集或基，且满足预算型约束：w(S)≤L 或 w(S)≥L，并且 opt∈{min,max}。P-MOL 为一大类 NP-困难优化问题提供了统一表示，包括带预算的拟阵独立集、约束最小基以及带拟阵约束的背包覆盖变体。此外，它自然扩展到多个拟阵约束。特别地，我们考虑拟阵交覆盖（MIC）问题，其中可行性由两个拟阵的公共独立集定义，目标是求最小 v(S) 满足 w(S)≥L。我们的主要结果是为所有非平凡的 P-MOL 变体提供一个统一的 EPTAS，该 EPTAS 通过推广 Hassin 和 Levin（SIAM J. Comput., 2004）用于解决约束最小生成树问题的技术而获得。具体而言，对于任意固定的 ε>0，我们提出一个运行时间为 |E|^{O(1)} (1/ε²)^{O(1/ε)} 的算法，该算法输出一个可行解 S，其值对于最小化变体至多为 (1+ε)OPT，对于最大化变体至少为 (1−ε)OPT。这解决了 P-MOL 所有成员的复杂性状态，因为这些问题均不存在 FPTAS（Doron-Arad, Kulik 和 Shachnai, ICALP'24）。最后，我们将 P-MOL 族与其到拟阵交的扩展区分开来。我们证明在覆盖约束下，P-MOL 的拟阵交变体不太可能存在 EPTAS，而在预算约束下已知存在 EPTAS。这凸显了这两种线性约束之间的定性差异，而这种差异在单拟阵设置中不会出现。
英文摘要
We study the following class of matroid optimization problems with a linear constraint (P-MOL). Given a matroid M=(E,I), two weight functions $v,w:E\to R_{\ge 0}$, and a threshold $L\in R_{\ge 0}$, find $opt v(S)$ where S is either an independent set or a base of M satisfying a budget-type constraint: $w(S)\le L$ or $w(S)\ge L$, and $opt\in\{min,max\}$. P-MOL provides a unified representation for a broad family of NP-hard optimization problems, including budgeted matroid independent set, constrained minimum-basis, and knapsack-cover variants with a matroid constraint. Also, it naturally extends to multiple matroid constraints. In particular, we consider the matroid intersection cover (MIC) problem, where feasibility is defined by the common independent sets of two matroids and one seeks minimum $v(S)$ subject to $w(S)\ge L$. Our main result is a unified EPTAS for all nontrivial P-MOL variants, obtained by generalizing a technique of Hassin and Levin (SIAM J. Comput., 2004) for solving the constrained minimum spanning tree problem. Specifically, for any fixed $ε>0$, we present an algorithm running in time $|E|^{O(1)} (1/{ε^2})^{O(1/ε)}$ that outputs a feasible solution S whose value is at most $(1+ε)OPT$ for minimization variants and at least $(1-ε)OPT$ for maximization variants. This resolves the complexity status of all members of P-MOL, as none of these problems admits an FPTAS (Doron-Arad, Kulik and Shachnai, ICALP'24). Finally, we separate the P-MOL family from its extension to matroid intersection. We show that an EPTAS is unlikely to exist for the matroid intersection variant of P-MOL under a covering constraint, whereas an EPTAS is known to exist under a budget constraint. This highlights a qualitative difference between these two types of linear constraints that does not arise in the single-matroid setting.
DOI：10.4230/LIPIcs.APPROX/RANDOM.2026.18
