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arXiv 1003.2357math.CO
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盒性与偏序集维数

Boxicity and Poset Dimension

Abhijin Adiga, Diptendu Bhowmick, L. Sunil Chandran

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摘要：

设$G$是一个简单的、无向的、有限的图，其顶点集为$V(G)$，边集为$E(G)$。$k$维盒是闭区间$[a_1,b_1]\times [a_2,b_2]\times...\times [a_k,b_k]$的笛卡尔积. $G$，$\boxi(G)$的{ \it boxicity}是最小整数$k$，使得$G$可以被表示为$k$维盒的相交图，即每个顶点被映射到$k$维盒，并且当且仅当它们对应的盒相交时，两个顶点在$G$中相邻.设$\poset=(S,P)$是偏序集，其中$S$是基集，$P$是$S$上的自反、反对称和传递的二元关系。$\poset$，$\dim(\poset)$的维数是最小整数$t$，使得$P$可以表示为$t$总阶数的交集。令$G_\poset$为$\poset$的\emph{underlying comparability graph}，即$S$为顶点集，且两个顶点相邻当且仅当它们在$\poset$中可比较。具有相同基础可比图的偏序集具有相同的维数是众所周知的事实。本文的第一个结果将偏序集的维数与其基础相似图的盒性联系起来。特别地，我们证明了对于任意偏序集$\poset$，$\boxi(G_\poset)/(χ(G_\poset)-1) \le \dim(\poset)\le 2\boxi(G_\poset)$，其中$χ(G_\poset)$是$G_\poset$和$χ(G_\poset)\ne1$的色数.紧接着，如果$\poset$是一个高度为2的偏序集，那么$\boxi(G_\poset)\le \dim(\poset)\le 2\boxi(G_\poset)$，因为高度为2的偏序集的底层可比图是一个二部图。本文的第二个结果将图$G$的盒性与$G$的\emph{extended double cover}相关的自然偏序联系起来，记为$G_c$：注意，$G_c$是具有部分集$A$和$B$的二分图，部分集$A$和$B$是$V(G)$的副本，使得对应于每个$u\in V(G)$，有两个顶点$u_A\in A$和$u_B\in B$，当且仅当$u=v$或$u$与$G$中的$v$相邻时，$\{u_A,v_B\}$是$G_c$中的边。令$\poset_c$是与$G_c$相关联的自然高度-2偏序集，通过使$A$为最小元素的集合，$B$为最大元素的集合。我们证明了$\frac{\boxi(G)}{2} \le \dim(\poset_c) \le 2\boxi(G)+4$ .这些结果有一些直接和重要的后果。上界$\dim(\poset)\le 2\boxi(G_\poset)$允许我们推导出迄今未知的偏序集维数的上界，如$\dim(\poset)\le 2\tw(G_\poset)+4$，因为已知任何图的盒性至多是它的$\tw+2$。在另一个方向上，利用已知的偏序维数的界，我们得到：（1）任何具有最大度$Δ$的图的盒性是$O(Δ\log^2Δ)$，这是对$Δ^2+2$的最佳已知上界的改进。(2)存在具有盒度$Ω(Δ\logΔ)$的图。这就否定了一个猜想，即图的盒性是$O(Δ)$。(3)对于任何$ε>0$，都不存在多项式时间算法来近似$n$顶点上的二分图的盒性，除非$NP=ZPP$。

英文摘要：

Let $G$ be a simple, undirected, finite graph with vertex set $V(G)$ and edge set $E(G)$. A $k$-dimensional box is a Cartesian product of closed intervals $[a_1,b_1]\times [a_2,b_2]\times...\times [a_k,b_k]$. The {\it boxicity} of $G$, $\boxi(G)$ is the minimum integer $k$ such that $G$ can be represented as the intersection graph of $k$-dimensional boxes, i.e. each vertex is mapped to a $k$-dimensional box and two vertices are adjacent in $G$ if and only if their corresponding boxes intersect. Let $\poset=(S,P)$ be a poset where $S$ is the ground set and $P$ is a reflexive, anti-symmetric and transitive binary relation on $S$. The dimension of $\poset$, $\dim(\poset)$ is the minimum integer $t$ such that $P$ can be expressed as the intersection of $t$ total orders. Let $G_\poset$ be the \emph{underlying comparability graph} of $\poset$, i.e. $S$ is the vertex set and two vertices are adjacent if and only if they are comparable in $\poset$. It is a well-known fact that posets with the same underlying comparability graph have the same dimension. The first result of this paper links the dimension of a poset to the boxicity of its underlying comparability graph. In particular, we show that for any poset $\poset$, $\boxi(G_\poset)/(χ(G_\poset)-1) \le \dim(\poset)\le 2\boxi(G_\poset)$, where $χ(G_\poset)$ is the chromatic number of $G_\poset$ and $χ(G_\poset)\ne1$. It immediately follows that if $\poset$ is a height-2 poset, then $\boxi(G_\poset)\le \dim(\poset)\le 2\boxi(G_\poset)$ since the underlying comparability graph of a height-2 poset is a bipartite graph. The second result of the paper relates the boxicity of a graph $G$ with a natural partial order associated with the \emph{extended double cover} of $G$, denoted as $G_c$: Note that $G_c$ is a bipartite graph with partite sets $A$ and $B$ which are copies of $V(G)$ such that corresponding to every $u\in V(G)$, there are two vertices $u_A\in A$ and $u_B\in B$ and $\{u_A,v_B\}$ is an edge in $G_c$ if and only if either $u=v$ or $u$ is adjacent to $v$ in $G$. Let $\poset_c$ be the natural height-2 poset associated with $G_c$ by making $A$ the set of minimal elements and $B$ the set of maximal elements. We show that $\frac{\boxi(G)}{2} \le \dim(\poset_c) \le 2\boxi(G)+4$. These results have some immediate and significant consequences. The upper bound $\dim(\poset)\le 2\boxi(G_\poset)$ allows us to derive hitherto unknown upper bounds for poset dimension such as $\dim(\poset)\le 2\tw(G_\poset)+4$, since boxicity of any graph is known to be at most its $\tw+2$. In the other direction, using the already known bounds for partial order dimension we get the following: (1) The boxicity of any graph with maximum degree $Δ$ is $O(Δ\log^2Δ)$ which is an improvement over the best known upper bound of $Δ^2+2$. (2) There exist graphs with boxicity $Ω(Δ\logΔ)$. This disproves a conjecture that the boxicity of a graph is $O(Δ)$. (3) There exists no polynomial-time algorithm to approximate the boxicity of a bipartite graph on $n$ vertices with a factor of $O(n^{0.5-ε})$ for any $ε>0$, unless $NP=ZPP$.

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