同质性的价格是多项式
The price of homogeneity is polynomial
AI总结：
本文通过提供多项式界，解决了同质墙引理中关于h(q,k)的复杂性问题，改进了图矿主理论中的算法效率。
摘要：
我们为同质墙引理提供了显式和多项式界，该引理首次隐含出现在Robertson和Seymour的图矿主系列第13项[JCTB 1990]中，并已成为图矿主理论算法理论的核心。一个墙中每个砖块被分配一组颜色被称为同质的，如果每个砖块分配相同的颜色集。同质墙引理指出，存在一个函数h，给定非负整数q和k以及一个h(q,k)-墙W，其中每个砖块被分配一个可能为空的{1,…,q}的子集，包含一个k-墙W'作为子图，使得如果将W'中每个砖块B分配给W中其内部砖块分配的集合的并集，则W'是同质的。众所周知，h(q,k)∈k^{O(q)}。同质墙引理在大多数应用中起关键作用，其中h的指数依赖性通常导致元参数的非均匀依赖，最坏情况下还会导致额外的指数膨胀。通过证明h(q,k)∈O(q^4·k^6)，我们回答了Sau、Stamoulis和Thilikos在ICALP 2020中提出的问题。
英文摘要：
We provide explicit and polynomial bounds for the Homogeneous Wall Lemma which occurred for the first time implicitly in the $13$th entry of Robertson and Seymour's Graph Minors Series [JCTB 1990] and has since become a cornerstone in the algorithmic theory of graph minors. A wall where each brick is assigned a set of colours is said to be homogeneous if each brick is assigned the same set of colours. The Homogeneous Wall Lemma says that there exists a function $h$ that, given non-negative integers $q$ and $k$ and an $h(q,k)$-wall $W$ where each brick is assigned a, possibly empty, subset of $\{ 1, \ldots , q \}$ contains a $k$-wall $W'$ as a subgraph such that, if one assigns to each brick $B$ of $W'$ the union of the sets assigned to the bricks of $W$ in its interior, then $W'$ is homogeneous. It is well-known that $h(q,k) \in k^{\mathcal{O}(q)}$. The Homogeneous Wall Lemma plays a key role in most applications of the Irrelevant Vertex Technique where an exponential dependency of $h$ on $q$ usually causes non-uniform dependencies on meta-parameters at best and additional exponential blow-ups at worst. By proving that $h(q,k) \in \mathcal{O}(q^4 \cdot k^6)$, we provide a positive answer to a problem raised by Sau, Stamoulis, and Thilikos [ICALP 2020].
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