修剪Moebius逆与有界度图
Trimmed Moebius Inversion and Graphs of Bounded Degree
摘要：
我们研究如何加快耶茨的算法计算zeta和Moebius变换的子集格上定义的函数。我们开发了一个修剪变形的Moebius反演，逐点进行，在考虑其超集之前完成一个子集的计算。对于一个$n$ -元素论域$U$和它的子集族$\scr F$，修剪Moebius反演允许我们计算$U$的填充、覆盖和划分的数量，其中$k$集在时间上在$\scr F$的成员的超集数量的多项式因子（在$n$中）内来自$\scr F$。依赖于一个相交定理的钟等。（1986），以限制集族的大小，我们将这些想法充分研究的组合优化问题的最大程度的图$Δ$。特别是，我们展示了如何计算在$(2^{Δ+1-2)^{n/(Δ+1)$的多项式因子内的时间的Domatic数和在$(2^{Δ+1-Δ-1)^{n/(Δ+1)$的多项式因子内的时间的色数。对于任何常数$Δ$，这些界限对于$ε>0$是$O\bigl((2-ε)^n\bigr)$，与顶点数$n$无关。
英文摘要：
We study ways to expedite Yates's algorithm for computing the zeta and Moebius transforms of a function defined on the subset lattice. We develop a trimmed variant of Moebius inversion that proceeds point by point, finishing the calculation at a subset before considering its supersets. For an $n$-element universe $U$ and a family $\scr F$ of its subsets, trimmed Moebius inversion allows us to compute the number of packings, coverings, and partitions of $U$ with $k$ sets from $\scr F$ in time within a polynomial factor (in $n$) of the number of supersets of the members of $\scr F$. Relying on an intersection theorem of Chung et al. (1986) to bound the sizes of set families, we apply these ideas to well-studied combinatorial optimisation problems on graphs of maximum degree $Δ$. In particular, we show how to compute the Domatic Number in time within a polynomial factor of $(2^{Δ+1-2)^{n/(Δ+1)$ and the Chromatic Number in time within a polynomial factor of $(2^{Δ+1-Δ-1)^{n/(Δ+1)$. For any constant $Δ$, these bounds are $O\bigl((2-ε)^n\bigr)$ for $ε>0$ independent of the number of vertices $n$.
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