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.
翻译:Let G - 2 = (V,w) 美元是一个带有 美元边缘值的加权非方向图形。 $G$的削减维度是 $G$最低削减量的特质矢量的维度, 以 $ 0, 1 美元为 美元为 。 对于每 美元= 2 美元为 美元, 美元为 美元= 美元 = (V, w) 美元为 美元 的加权非方向图 。 削减维度最近由 Graur 和 al.\\ cite{GPRW20} 来定义。 他显示 美元最低削减量为 美元 的特质矢量最大削减维度的维度, 美元为 美元 - 美元为 美元 最低量。 我们观察到, 确定量为 美元 美元 最低量为 美元 最低量 的离值 。