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 or cycles, we determine the extremal graphs G with the minimum number of edges such that VC P (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.
[1]
Vladimir Vapnik,et al.
Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities
,
1971
.
[2]
Mihalis Yannakakis,et al.
On limited nondeterminism and the complexity of the V-C dimension
,
1993,
[1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.
[3]
David S. Johnson,et al.
Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran
,
1979
.
[4]
Bernard Chazelle,et al.
Quasi-optimal range searching in spaces of finite VC-dimension
,
1989,
Discret. Comput. Geom..
[5]
David S. Johnson,et al.
The NP-Completeness Column: An Ongoing Guide
,
1982,
J. Algorithms.
[6]
David Haussler,et al.
Learnability and the Vapnik-Chervonenkis dimension
,
1989,
JACM.
[7]
Mihalis Yannakakis,et al.
Node-and edge-deletion NP-complete problems
,
1978,
STOC.
[8]
David Haussler,et al.
Epsilon-nets and simplex range queries
,
1986,
SCG '86.