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# On the Cut Dimension of a Graph

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LIPIcs.CCC.2021.15.pdf
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## Cite As

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

## 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.

## Subject Classification

##### 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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## References

1. Sepehr Assadi, Deeparnab Chakrabarty, and Sanjeev Khanna. Graph connectivity and single element recovery via linear and OR queries. CoRR, abs/2007.06098, 2020. URL: http://arxiv.org/abs/2007.06098.
2. László Babai, Peter Frankl, and Janos Simon. Complexity classes in communication complexity theory (preliminary version). In 27th Annual Symposium on Foundations of Computer Science, Toronto, Canada, 27-29 October 1986, pages 337-347, 1986. URL: https://doi.org/10.1109/SFCS.1986.15.
3. András A. Benczúr and Michel X. Goemans. Deformable polygon representations and near-mincuts. In Martin Grøtschel and Gyula O. H. Katona, editors, Building Bridges: Between Mathematics and Computer Science, volume 19 of Bolyai Society Mathematical Studies, pages 103-135. Springer, 2008.
4. R. E. Bixby. The minimum number of edges and vertices in a graph with edge connectivity n and m n-bonds. Netw., 5(3):253–298, 1975. URL: https://doi.org/10.1002/net.1975.5.3.253.
5. L. Sunil Chandran and L. Shankar Ram. On the number of minimum cuts in a graph. SIAM J. Discret. Math., 18(1):177-194, 2004. URL: https://doi.org/10.1137/S0895480103427138.
6. Efim A. Dinitz, Alexander V. Karzanov, and Michael V. Lomonosov. On the structure of the system of minimum edge cuts of a graph. Studies in discrete optimization, 1976.
7. Tamás Fleiner and András Frank. A quick proof for the cactus representation of mincuts. EGRES Quick Proof, 2009-03, 2009.
8. Pawel Gawrychowski, Shay Mozes, and Oren Weimann. Minimum cut in O(m log² n) time. In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors, 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference), volume 168 of LIPIcs, pages 57:1-57:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.57.
9. Pawel Gawrychowski, Shay Mozes, and Oren Weimann. A note on a recent algorithm for minimum cut. In Hung Viet Le and Valerie King, editors, 4th Symposium on Simplicity in Algorithms, SOSA 2021, Virtual Conference, January 11-12, 2021, pages 74-79. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976496.8.
10. Mohsen Ghaffari, Krzysztof Nowicki, and Mikkel Thorup. Faster algorithms for edge connectivity via random 2-out contractions. In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 1260-1279. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.77.
11. Michel X. Goemans. Minimum bounded degree spanning trees. In 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), 21-24 October 2006, Berkeley, California, USA, Proceedings, pages 273-282. IEEE Computer Society, 2006. URL: https://doi.org/10.1109/FOCS.2006.48.
12. Michel X. Goemans and V. S. Ramakrishnan. Minimizing submodular functions over families of sets. Comb., 15(4):499-513, 1995. URL: https://doi.org/10.1007/BF01192523.
13. Andrei Graur, Tristan Pollner, Vidhya Ramaswamy, and S. Matthew Weinberg. New query lower bounds for submodular function minimization. In Thomas Vidick, editor, 11th Innovations in Theoretical Computer Science Conference, ITCS 2020, January 12-14, 2020, Seattle, Washington, USA, volume 151 of LIPIcs, pages 64:1-64:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.64.
14. András Hajnal, Wolfgang Maass, and György Turán. On the communication complexity of graph properties. In Proceedings of the 20th Annual ACM Symposium on Theory of Computing, May 2-4, 1988, Chicago, Illinois, USA, pages 186-191, 1988. URL: https://doi.org/10.1145/62212.62228.
15. Nicholas J. A. Harvey. Matchings, matroids and submodular functions. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, 2008. URL: http://hdl.handle.net/1721.1/44416.
16. Monika Henzinger, Satish Rao, and Di Wang. Local flow partitioning for faster edge connectivity. SIAM J. Comput., 49(1):1-36, 2020. URL: https://doi.org/10.1137/18M1180335.
17. Monika Rauch Henzinger and David P. Williamson. On the number of small cuts in a graph. Inf. Process. Lett., 59(1):41-44, 1996. URL: https://doi.org/10.1016/0020-0190(96)00079-8.
18. Kamal Jain. A factor 2 approximation algorithm for the generalized Steiner network problem. Comb., 21(1):39-60, 2001. URL: https://doi.org/10.1007/s004930170004.
19. David R. Karger. Global min-cuts in RNC, and other ramifications of a simple min-cut algorithm. In Vijaya Ramachandran, editor, Proceedings of the Fourth Annual ACM/SIGACT-SIAM Symposium on Discrete Algorithms, 25-27 January 1993, Austin, Texas, USA, pages 21-30. ACM/SIAM, 1993. URL: http://dl.acm.org/citation.cfm?id=313559.313605.
20. David R. Karger. Minimum cuts in near-linear time. J. ACM, 47(1):46-76, 2000. URL: https://doi.org/10.1145/331605.331608.
21. Ken-ichi Kawarabayashi and Mikkel Thorup. Deterministic edge connectivity in near-linear time. J. ACM, 66(1):4:1-4:50, 2019. URL: https://doi.org/10.1145/3274663.
22. Bernhard Korte and Jens Vygen. Combinatorial Optimization: Theory and Algorithms. Springer, 2018.
23. László Lovász. Combinatorial problems and exercises (2. ed.). North-Holland, 1993.
24. Sagnik Mukhopadhyay and Danupon Nanongkai. Weighted min-cut: sequential, cut-query, and streaming algorithms. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 496-509, 2020.
25. Hiroshi Nagamochi, Kazuhiro Nishimura, and Toshihide Ibaraki. Computing all small cuts in an undirected network. SIAM J. Discret. Math., 10(3):469-481, 1997. URL: https://doi.org/10.1137/S0895480194271323.
26. Aviad Rubinstein, Tselil Schramm, and S. Matthew Weinberg. Computing exact minimum cuts without knowing the graph. In 9th Innovations in Theoretical Computer Science Conference, ITCS 2018, January 11-14, 2018, Cambridge, MA, USA, pages 39:1-39:16, 2018. URL: https://doi.org/10.4230/LIPIcs.ITCS.2018.39.
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