Abstract
We study set systems over the vertex set (or edge set) of some graph that are induced by special graph properties like clique, connectedness, path, star, tree, etc. We derive a variety of combinatorial and computational results on the VC (Vapnik-Chervonenkis) dimension of these set systems.
For most of these set systems (e.g. for the systems induced by trees, connected sets, or paths), computing the VC-dimension is an NP-hard problem. Moreover, determining the VC-dimension for set systems induced by neighborhoods of single vertices is complete for the class LogNP. In contrast to these intractability results, we show that the VC-dimension for set systems induced by stars is computable in polynomial time. For set systems induced by paths, we determine the extremal graphs G with the minimum number of edges such that VCp(G)≥k. Finally, we show a close relation between the VC-dimension of set systems induced by connected sets of vertices and the VC dimension of set systems induced by connected sets of edges; the argument is done via the line graph of the corresponding graph.
Research supported in part by NSERC (National Science and Engineering Research Council of Canada) grant.
Research supported in part by a grant from the DAAD (German Academic Exchange Service).
Research supported by the Spezialforschungsbereich F 003 “Optimierung und Kontrolle”;, Projektbereich Diskrete Optimierung.
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References
M. Anthony, G. Brightwell, and C. Cooper, “On the Vapnik-Chervonenkis Dimension of a Graph”;, Technical Report, London School of Economics, 1993.
A. Blumer, A. Ehrenfeucht, D. Haussler, and M. K. Warmuth, “Learnability and the Vapnik-Chervonenkis Dimension”;, Journal of the ACM, 36: 929–965, 1989.
B. Bollobás, “Extremal Graph Theory”;, Academic Press, London, 1978.
B. Chazelle and E. Welzl, “Quasi-Optimal Range Searching and VC-Dimensions”;, Discrete & Computational Geometry, 4: 467–490, 1989.
M. R. Garey and D. S. Johnson, “Computers and intractability”; Freeman, San Francisco, 1979.
D. Haussler and E. Welzl, “Epsilon-nets and Simplex Range Queries”;, Discrete & Computational Geometry, 2: 127–151, 1987.
R. L. Hemminger and L. W. Beineke, “Line Graphs and Line Digraphs”;, in: L.W. Beineke and R.J. Wilson, editors, Selected topics in graph theory, pages 271–305, Academic Press, London, 1978.
D. S. Johnson, “The NP-completeness column: An ongoing guide”;, Journal of Algorithms, 8: 285–303 (1987).
E. Kranakis, D. Krizanc, B. Ruf, J. Urrutia and G. Wöginger, “VC-Dimensions for Graphs”;, Technical Report SCS-TR-255, Carleton University, School of Computer Science, Ottawa, 1994.
C. H. Papadimitriou and M. Yannakakis, “On limited nondeterminism and the complexity of VC-dimension”;, Proceedings of the eighth annual Conference on Structure in Complexity Theory, 12–18, IEEE, 1993.
H.-I. Lu and R. Ravi, “The Power of Local Optimization: Approximation Algorithms for Maximum-Leaf Spanning Tree”;, Proceedings of 1992 Allerton Conference.
P. Turán, “On an extremal problem in graph theory”;, Mat. Fiz. Lapok, 48: 436–452, 1941 (in Hungarian).
V. N. Vapnik and A. Ya. Chervonenkis, “On the Uniform Convergence of Relative Frequencies of Events to their Probabilities”;, Theory of Probability and its Applications, 16(2): 264–280, 1971.
M. Yannakakis, “Node-and Edge-deletion NP-complete Problems”;, Proc. 10th Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, New York, 253–264, 1978.
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© 1995 Springer-Verlag Berlin Heidelberg
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Kranakis, E., Krizanc, D., Ruf, B., Urrutia, J., Woeginger, G.J. (1995). VC-dimensions for graphs (extended abstract). In: Nagl, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 1995. Lecture Notes in Computer Science, vol 1017. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60618-1_61
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DOI: https://doi.org/10.1007/3-540-60618-1_61
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