Abstract | ||
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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. |
Year | DOI | Venue |
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2021 | 10.4230/LIPIcs.CCC.2021.15 | CCC |
DocType | Citations | PageRank |
Conference | 0 | 0.34 |
References | Authors | |
0 | 4 |
Name | Order | Citations | PageRank |
---|---|---|---|
Troy Lee | 1 | 276 | 28.96 |
Tongyang Li | 2 | 17 | 6.26 |
Miklos Santha | 3 | 728 | 92.42 |
Shengyu Zhang | 4 | 329 | 42.48 |