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On the Cut Dimension of a Graph

Authors Troy Lee, Tongyang Li, Miklos Santha, Shengyu Zhang

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ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Query complexity
  • submodular function minimization
  • cut dimension

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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.

Cite As Get BibTex

Troy Lee, Tongyang Li, Miklos Santha, and Shengyu Zhang. On the Cut Dimension of a Graph. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 15:1-15:35, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.CCC.2021.15

Author Details

Troy Lee
  • Centre for Quantum Software and Information, University of Technology Sydney, Australia
Tongyang Li
  • Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA, USA
  • Center on Frontiers of Computing Studies, Peking University, Beijing, China
Miklos Santha
  • CNRS, IRIF, Université de Paris, France
  • Centre for Quantum Technologies and MajuLab, National University of Singapore, Singapore
Shengyu Zhang
  • Tencent Quantum Laboratory, Shenzhen, China

Funding

Research at CQT is funded by the National Research Foundation, the Prime Minister’s Office, and the Ministry of Education, Singapore under the Research Centres of Excellence programme’s research grant R-710-000-012-135. In addition, this work has been supported in part by the QuantERA ERA-NET Cofund project QuantAlgo and the ANR project ANR-18-CE47-0010 QUDATA.
  • Lee, Troy: Troy Lee is supported in part by the Australian Research Council Grant No: DP200100950.
  • Li, Tongyang: Tongyang Li is supported by the ARO contract W911NF-17-1-0433, NSF grant PHY-1818914, and an NSF QISE-NET Triplet Award (grant DMR-1747426).

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