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### On the Cut Dimension of a Graph

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### Abstract

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 ≥ 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. [Andrei Graur et al., 2020], 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 ≥ 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 linear queries needed by a deterministic algorithm to solve mincut, where a linear query can ask any vector x ∈ 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 𝓁₁-approximate cut dimension. The 𝓁₁-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 𝓁₁-approximate cut dimension 2n-2, showing that it can be strictly larger than the cut dimension.

### BibTeX - Entry

```@InProceedings{lee_et_al:LIPIcs.CCC.2021.15,
author =	{Lee, Troy and Li, Tongyang and Santha, Miklos and Zhang, Shengyu},
title =	{{On the Cut Dimension of a Graph}},
booktitle =	{36th Computational Complexity Conference (CCC 2021)},
pages =	{15:1--15:35},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-193-1},
ISSN =	{1868-8969},
year =	{2021},
volume =	{200},
editor =	{Kabanets, Valentine},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
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