VC-dimensions for graphs (extended 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.
KeywordsPolynomial Time Span Tree Line Graph Graph Property Extremal Graph
Unable to display preview. Download preview PDF.
- 1.M. Anthony, G. Brightwell, and C. Cooper, “On the Vapnik-Chervonenkis Dimension of a Graph”;, Technical Report, London School of Economics, 1993.Google Scholar
- 3.B. Bollobás, “Extremal Graph Theory”;, Academic Press, London, 1978.Google Scholar
- 4.B. Chazelle and E. Welzl, “Quasi-Optimal Range Searching and VC-Dimensions”;, Discrete & Computational Geometry, 4: 467–490, 1989.Google Scholar
- 5.M. R. Garey and D. S. Johnson, “Computers and intractability”; Freeman, San Francisco, 1979.Google Scholar
- 6.D. Haussler and E. Welzl, “Epsilon-nets and Simplex Range Queries”;, Discrete & Computational Geometry, 2: 127–151, 1987.Google Scholar
- 7.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.Google Scholar
- 8.D. S. Johnson, “The NP-completeness column: An ongoing guide”;, Journal of Algorithms, 8: 285–303 (1987).Google Scholar
- 9.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.Google Scholar
- 10.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.Google Scholar
- 11.H.-I. Lu and R. Ravi, “The Power of Local Optimization: Approximation Algorithms for Maximum-Leaf Spanning Tree”;, Proceedings of 1992 Allerton Conference.Google Scholar
- 12.P. Turán, “On an extremal problem in graph theory”;, Mat. Fiz. Lapok, 48: 436–452, 1941 (in Hungarian).Google Scholar
- 13.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.Google Scholar
- 14.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.Google Scholar