We study how good a lexicographically maximal solution is in the weighted matching and matroid intersection problems. A solution is lexicographically maximal if it takes as many heaviest elements as possible, and subject to this, it takes as many second heaviest elements as possible, and so on. If the distinct weight values are sufficiently dispersed, e.g., the minimum ratio of two distinct weight values is at least the ground set size, then the lexicographical maximality and the usual weighted optimality are equivalent. We show that the threshold of the ratio for this equivalence to hold is exactly $2$. Furthermore, we prove that if the ratio is less than $2$, say $\alpha$, then a lexicographically maximal solution achieves $(\alpha/2)$-approximation, and this bound is tight.

中文翻译：

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匹配和拟阵交问题中字典序极大解的逼近

我们研究了字典序最大解在加权匹配和拟阵相交问题中有多好。如果一个解采用尽可能多的最重元素，那么它在字典上是最大的，并且在此基础上，它需要尽可能多的第二重元素，依此类推。如果不同的权重值足够分散，例如，两个不同的权重值的最小比率至少是基础集大小，那么词典最大和通常的加权最优是等效的。我们表明，保持这种等价性的比率阈值恰好是 2 美元。此外，我们证明，如果该比率小于 $2$，例如 $\alpha$，则字典序最大解达到 $(\alpha/2)$-近似，并且这个界限很紧。