Let $G = (V,w)$ be a weighted undirected graph with $m$ edges. The cut dimension of $G$ is the dimension of the span of the characteristic vectors of the minimum cuts of $G$, viewed as vectors in $\{0,1\}^m$. For every $n \ge 2$ we show that the cut dimension of an $n$-vertex graph is at most $2n-3$, and construct graphs realizing this bound. The cut dimension was recently defined by Graur et al.\ \cite{GPRW20}, who show that the maximum cut dimension of an $n$-vertex graph is a lower bound on the number of cut queries needed by a deterministic algorithm to solve the minimum cut problem on $n$-vertex graphs. For every $n\ge 2$, Graur et al.\ exhibit a graph on $n$ vertices with cut dimension at least $3n/2 -2$, giving the first lower bound larger than $n$ on the deterministic cut query complexity of computing mincut. We observe that the cut dimension is even a lower bound on the number of \emph{linear} queries needed by a deterministic algorithm to solve mincut, where a linear query can ask any vector $x \in \mathbb{R}^{\binom{n}{2}}$ and receives the answer $w^T x$. Our results thus show a lower bound of $2n-3$ on the number of linear queries needed by a deterministic algorithm to solve minimum cut on $n$-vertex graphs, and imply that one cannot show a lower bound larger than this via the cut dimension. We further introduce a generalization of the cut dimension which we call the $\ell_1$-approximate cut dimension. The $\ell_1$-approximate cut dimension is also a lower bound on the number of linear queries needed by a deterministic algorithm to compute minimum cut. It is always at least as large as the cut dimension, and we construct an infinite family of graphs on $n=3k+1$ vertices with $\ell_1$-approximate cut dimension $2n-2$, showing that it can be strictly larger than the cut dimension.

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关于图的切割维度

令 $G = (V,w)$ 是一个带 $m$ 边的加权无向图。$G$的割维是$G$最小割的特征向量的跨度维数，视为$\{0,1\}^m$中的向量。对于每个$n\ge 2$，我们证明$n$-顶点图的切割维度至多为$2n-3$，并构造实现该界限的图。Graur 等人最近定义了切割维度。\ cite{GPRW20}，他们表明 $n$-顶点图的最大切割维度是确定性算法求解所需的切割查询数量的下限$n$-顶点图上的最小割问题。对于每个 $n\ge 2$，Graur 等人展示了一个关于 $n$ 顶点的图，切割维度至少为 $3n/2 -2$，给出了确定性切割查询上大于 $n$ 的第一个下限计算mincut的复杂度。我们观察到切割维度甚至是确定性算法解决 mincut 所需的 \emph{linear} 查询数量的下限，其中线性查询可以询问任何向量 $x \in \mathbb{R}^{\ binom{n}{2}}$ 并收到答案 $w^T x$。因此，我们的结果显示确定性算法解决 $n$-顶点图上的最小割所需的线性查询数量的下限为 $2n-3$，并暗示不能通过以下方式显示比此更大的下限切割尺寸。我们进一步介绍了切割维度的泛化，我们称之为 $\ell_1$-approximate cut 维度。$\ell_1$-approximate cut 维度也是确定性算法计算最小切割所需的线性查询数量的下限。它始终至少与切割尺寸一样大，