License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2021.15
URN: urn:nbn:de:0030-drops-142890
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14289/
Lee, Troy ;
Li, Tongyang ;
Santha, Miklos ;
Zhang, Shengyu
On the Cut Dimension of a Graph
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 ?₁-approximate cut dimension. The ?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 ?₁-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},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14289},
URN = {urn:nbn:de:0030-drops-142890},
doi = {10.4230/LIPIcs.CCC.2021.15},
annote = {Keywords: Query complexity, submodular function minimization, cut dimension}
}
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Keywords: |
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Query complexity, submodular function minimization, cut dimension |
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Collection: |
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36th Computational Complexity Conference (CCC 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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08.07.2021 |
