Document

# Practical Computation of Graph VC-Dimension

## File

LIPIcs.SEA.2024.8.pdf
• Filesize: 0.98 MB
• 20 pages

## Cite As

David Coudert, Mónika Csikós, Guillaume Ducoffe, and Laurent Viennot. Practical Computation of Graph VC-Dimension. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SEA.2024.8

## Abstract

For any set system ℋ = (V,ℛ), ℛ ⊆ 2^V, a subset S ⊆ V is called shattered if every S' ⊆ S results from the intersection of S with some set in ℛ. The VC-dimension of ℋ is the size of a largest shattered set in V. In this paper, we focus on the problem of computing the VC-dimension of graphs. In particular, given a graph G = (V,E), the VC-dimension of G is defined as the VC-dimension of (V, N), where N contains each subset of V that can be obtained as the closed neighborhood of some vertex v ∈ V in G. Our main contribution is an algorithm for computing the VC-dimension of any graph, whose effectiveness is shown through experiments on various types of practical graphs, including graphs with millions of vertices. A key aspect of its efficiency resides in the fact that practical graphs have small VC-dimension, up to 8 in our experiments. As a side-product, we present several new bounds relating the graph VC-dimension to other classical graph theoretical notions. We also establish the W[1]-hardness of the graph VC-dimension problem by extending a previous result for arbitrary set systems.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Graph algorithms analysis
• Theory of computation → Parameterized complexity and exact algorithms
• Theory of computation → Algorithm design techniques
• VC-dimension
• graph
• algorithm

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Martin Anthony, Graham Brightwell, and Colin Cooper. The Vapnik-Chervonenkis dimension of a random graph. Discrete Mathematics, 138(1-3):43-56, 1995. URL: https://doi.org/10.1016/0012-365X(94)00187-N.
2. Anselm Blumer, Andrzej Ehrenfeucht, David Haussler, and Manfred K. Warmuth. Learnability and the Vapnik-Chervonenkis dimension. Journal of the ACM (JACM), 36(4):929-965, 1989. URL: https://doi.org/10.1145/76359.76371.
3. Marthe Bonamy, Édouard Bonnet, Nicolas Bousquet, Pierre Charbit, Panos Giannopoulos, Eun Jung Kim, Paweł Rzażewski, Florian Sikora, and Stéphan Thomassé. EPTAS and subexponential algorithm for maximum clique on disk and unit ball graphs. Journal of the ACM (JACM), 68(2):1-38, 2021. URL: https://doi.org/10.1145/3433160.
4. Nicolas Bousquet, Aurélie Lagoutte, Zhentao Li, Aline Parreau, and Stéphan Thomassé. Identifying codes in hereditary classes of graphs and VC-dimension. SIAM Journal on Discrete Mathematics, 29(4):2047-2064, 2015. URL: https://doi.org/10.1137/14097879.
5. Jérémie Chalopin, Victor Chepoi, Fionn Mc Inerney, Sébastien Ratel, and Yann Vaxès. Sample compression schemes for balls in graphs. SIAM Journal on Discrete Mathematics, 37(4):2585-2616, 2023.
6. Jérémie Chalopin, Victor Chepoi, Shay Moran, and Manfred K. Warmuth. Unlabeled sample compression schemes and corner peelings for ample and maximum classes. Journal of Computer and System Sciences, 127:1-28, 2022. URL: https://doi.org/10.1016/j.jcss.2022.01.003.
7. Bernard Chazelle and Emo Welzl. Quasi-optimal range searching in spaces of finite VC-dimension. Discrete & Computational Geometry, 4:467-489, 1989. URL: https://doi.org/10.1007/BF02187743.
8. Victor Chepoi, Bertrand Estellon, and Yann Vaxes. Covering planar graphs with a fixed number of balls. Discrete & Computational Geometry, 37:237-244, 2007. URL: https://doi.org/10.1007/s00454-006-1260-0.
9. Victor Chepoi, Arnaud Labourel, and Sébastien Ratel. On density of subgraphs of Cartesian products. Journal of Graph Theory, 93(1):64-87, 2020. URL: https://doi.org/10.1002/jgt.22469.
10. David Coudert, Mónika Csikós, Guillaume Ducoffe, and Laurent Viennot. Graph VC-dimension. Software, swhId: https://archive.softwareheritage.org/swh:1:dir:2edb922280298023cb1cc66ff7e5dd76b3a489b8;origin=https://gitlab.inria.fr/viennot/graph-vcdim;visit=swh:1:snp:e559bbd07826dec77e48c9eb10eac1a459099c55;anchor=swh:1:rev:4dab92800397934bb5dc003e2965fd372c2fa50b (visited on 2024-06-28). URL: https://gitlab.inria.fr/viennot/graph-vcdim.
11. Mónika Csikós and Nabil H. Mustafa. Optimal approximations made easy. Information Processing Letters, 176:106250, 2022. URL: https://doi.org/10.1016/j.ipl.2022.106250.
12. Erik D. Demaine, Felix Reidl, Peter Rossmanith, Fernando Sánchez Villaamil, Somnath Sikdar, and Blair D. Sullivan. Structural sparsity of complex networks: Bounded expansion in random models and real-world graphs. Journal of Computer and System Sciences, 105:199-241, 2019. URL: https://doi.org/10.1016/j.jcss.2019.05.004.
13. Christian J. J. Despres. The Vapnik-Chervonenkis dimension of cubes in ℝ^d, 2017. URL: https://arxiv.org/abs/1412.6612.
14. Rodney G. Downey, Patricia A. Evans, and Michael R. Fellows. Parameterized learning complexity. In Proceedings of the sixth annual conference on Computational learning theory, pages 51-57, 1993.
15. Guillaume Ducoffe. On computing the average distance for some chordal-like graphs. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS), 2021.
16. Guillaume Ducoffe. The diameter of AT-free graphs. Journal of Graph Theory, 99(4):594-614, 2022. URL: https://doi.org/10.1002/jgt.22754.
17. Guillaume Ducoffe, Michel Habib, and Laurent Viennot. Diameter, eccentricities and distance oracle computations on H-minor free graphs and graphs of bounded (distance) Vapnik-Chervonenkis dimension. SIAM Journal on Computing, 51(5):1506-1534, 2022. URL: https://doi.org/10.1137/20M136551.
18. Sally Floyd and Manfred Warmuth. Sample compression, learnability, and the Vapnik-Chervonenkis dimension. Machine learning, 21:269-304, 1995. URL: https://doi.org/10.1023/A:1022660318680.
19. Jacob Fox, János Pach, and Andrew Suk. Bounded VC-dimension implies the Schur-Erdős conjecture. Combinatorica, 41(6):803-813, 2021. URL: https://doi.org/10.1007/s00493-021-4530-9.
20. Michel Habib, Ross McConnell, Christophe Paul, and Laurent Viennot. Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing. Theoretical Computer Science, 234(1-2):59-84, 2000. URL: https://doi.org/10.1016/S0304-3975(97)00241-7.
21. David Haussler and Emo Welzl. Epsilon-nets and simplex range queries. In Proceedings of the second annual symposium on Computational geometry, pages 61-71, 1986.
22. Sean B. Holden and Mahesan Niranjan. On the practical applicability of VC-dimension bounds. Neural Computation, 7(6):1265-1288, 1995. URL: https://doi.org/10.1162/neco.1995.7.6.1265.
23. Evangelos Kranakis, Danny Krizanc, Berthold Ruf, Jorge Urrutia, and Gerhard Woeginger. The VC-dimension of set systems defined by graphs. Discrete Applied Mathematics, 77(3):237-257, 1997. URL: https://doi.org/10.1016/S0166-218X(96)00137-0.
24. Jure Leskovec, Jon Kleinberg, and Christos Faloutsos. Graph evolution: Densification and shrinking diameters. ACM transactions on Knowledge Discovery from Data - TKDD, 1(1):2-42, 2007. URL: https://doi.org/10.1145/1217299.1217301.
25. Yi Li, Philip M. Long, and Aravind Srinivasan. Improved Bounds on the Sample Complexity of Learning. Journal of Computer and System Sciences, 62(3):516-527, 2001. URL: https://doi.org/10.1006/jcss.2000.1741.
26. Tomasz Łuczak and Stéphan Thomassé. Coloring dense graphs via VC-dimension. arXiv preprint arXiv:1007.1670, 2010.
27. Pasin Manurangsi and Aviad Rubinstein. Inapproximability of VC-dimension and Littlestone’s dimension. In Conference on Learning Theory, pages 1432-1460. PMLR, 2017.
28. Jiří Matoušek. Geometric set systems. In European Congress of Mathematics: Budapest, July 22-26, 1996 Volume II, pages 1-27. Springer, 1998.
29. Jiří Matoušek. VC-Dimension and Discrepancy, pages 137-169. Springer Berlin Heidelberg, 1999. URL: https://doi.org/10.1007/978-3-642-03942-3_5.
30. Rose Oughtred, Chris Stark, Bobby-Joe Breitkreutz, Jennifer Rust, Lorrie Boucher, Christie Chang, Nadine Kolas, Lara O’Donnell, Genie Leung, Rochelle McAdam, et al. The BioGRID interaction database: 2019 update. Nucleic acids research, 47(D1):D529-D541, 2019.
31. Robert Paige and Robert Endre Tarjan. Three partition refinement algorithms. SIAM Journal on Computing, 16(6):973-989, 1987. URL: https://doi.org/10.1137/0216062.
32. Christos H. Papadimitriou and Mihalis Yannakakis. On limited nondeterminism and the complexity of the VC dimension. Journal of Computer and System Sciences, 53(2):161-170, 1996. URL: https://doi.org/10.1006/jcss.1996.0058.
33. Lukasz Salwinski, Christopher S. Miller, Adam J. Smith, Frank K. Pettit, James U. Bowie, and David Eisenberg. The database of interacting proteins: 2004 update. Nucleic acids research, 32(suppl_1):D449-D451, 2004.
34. Yuval Shavitt and Eran Shir. DIMES: Let the internet measure itself. ACM SIGCOMM Computer Communication Review, 35(5):71-74, October 2005. URL: https://doi.org/10.1145/1096536.1096546.
35. The Cooperative Association for Internet Data Analysis (CAIDA). The CAIDA AS relationships dataset. http://www.caida.org/data/active/as-relationships/, 2013.
36. Vladimir N. Vapnik and Alexey Ya. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability & Its Applications, 16(2):264-280, 1971. URL: https://doi.org/10.1137/1116025.
37. Emo Welzl. Partition trees for triangle counting and other range searching problems. In Proceedings of the fourth annual symposium on Computational geometry, pages 23-33, 1988.
38. Roberta S Wenocur and Richard M Dudley. Some special vapnik-chervonenkis classes. Discrete Mathematics, 33(3):313-318, 1981. URL: https://doi.org/10.1016/0012-365X(81)90274-0.