AI 中文总结
本文研究图拟阵之外join最大规模μ(M)与耳分解参数η(M)的等式关系，证明该等式在余图拟阵等类中不成立、相关计算为NP难，同时给出多类拟阵的两参数定量比较界，包括正则拟阵的η(M)≤6μ(M)-2。
AI 中文摘要
对于拟阵$M$，join是指集合$J\subseteq E(M)$，其与每个回路$C$的交集元素个数至多为$|C|/2$。令$μ(M)$表示最大join的规模。受Frank针对图拟阵的最小-最大定理启发，我们将$μ(M)$与耳分解参数$η(M)=(r(M)+φ(M))/2$进行比较，其中$φ(M)$是$M$的耳分解中偶瓣的最小数量。Frank的定理表明，对于连通图拟阵，$μ(M)=η(M)$。\n 本文研究该等式在图拟阵之外的适用范围。我们证明该精确等式通常不成立：它在余图拟阵中就已不成立，因此在二元拟阵类中也不成立。此外，满足$μ(M)=η(M)$的拟阵类不是子式封闭的，因此几乎不可能通过禁用子式来刻画。我们还证明，对于余图拟阵，计算最大join是NP难的，近似比难以达到$519/520$以内；对于以基列表形式给出的稀疏铺砌拟阵，该问题也是NP难的。尽管存在这些负面结果，我们仍证明这两个参数在若干自然拟阵类中仍具有定量可比性。我们证明了二元拟阵、铺砌拟阵、余图拟阵以及任意连通拟阵的比较界。特别地，利用Seymour分解定理，我们结合图拟阵的等式结果、余图拟阵的界以及对$R_{10}$的直接分析，得到对于每个正则拟阵$M$，都有$η(M)\leq 6μ(M)-2$。
英文摘要
For a matroid $M$, a join is a set $J\subseteq E(M)$ that meets every circuit $C$ in at most $|C|/2$ elements. Let $μ(M)$ denote the maximum size of a join. Motivated by Frank's min--max theorem for graphic matroids, we compare $μ(M)$ with an ear-decomposition parameter $η(M)=(r(M)+φ(M))/2$, where $φ(M)$ is the minimum number of even lobes in an ear decomposition of $M$. Frank's theorem implies $μ(M)=η(M)$ for connected graphic matroids. Here we study how far this equality extends beyond graphic matroids. We show that the exact equality does not hold in general: it already fails for cographic matroids, hence within the binary class. Furthermore, the class of matroids satisfying $μ(M)=η(M)$ is not minor-closed, thus there is little hope for a forbidden minor characterization. We also prove that computing a maximum join is NP-hard for cographic matroids, hard to approximate within a factor of $519/520$, and NP-hard for sparse paving matroids given by their list of bases. Despite these negative results, we show that the two parameters remain quantitatively comparable in several natural classes. We prove comparison bounds for binary, paving, cographic, and arbitrary connected matroids. In particular, using Seymour's decomposition theorem, we combine the equality for graphic matroids, the bound for cographic matroids, and a direct analysis of $R_{10}$ to obtain $η(M)\leq 6μ(M)-2$ for every regular matroid $M$.
