Abstract
We present an application of the topological approach of Kahn, Saks and Sturtevant to the evasiveness conjecture for monotone graph properties. Although they proved evasiveness for every prime power of vertices, the best asymtotic lower bound for the (decision tree) complexity c(n) known so far is ¼n 2, proved in the same paper. In case that the evasiveness conjecture holds, it is ½n(n−1).We demonstrate some techniques to improve the 1/4 bound and show \( c(n) \geqslant \tfrac{8} {{25}}n^2 - o(n^2 ) \).
Similar content being viewed by others
References
A. Björner: Topological methods, in: Handbook of Combinatorics (Graham et al., eds.), Vol. 2, 1995 ([5]), pages 1819–1872.
M. R. Best, P. van Emde Boas and H. W. Lenstra Jr.: A sharpened version of the Aanderaa-Rosenberg conjecture, Math. Centrum Tracts Report ZW 30/74, Amsterdam, 1974.
A. Chakrabarti, S. Khot and Y. Shi: Evasiveness of subgraph containment and related properties, SIAM J. Comput. 31(3) (2001), 866–875.
J. D. Dixon and B. Mortimer: Permutation Groups, volume 163, Springer-Verlag, 1996.
R. L. Graham, M. Grötschel and L. Lovász, editors: Handbook of combinatorics, Vol. 2, Cambridge, MA, USA, 1995, MIT Press.
J. Kahn, M. Saks and D. Sturtevant: A topological approach to evasiveness, Combinatorica4(4) (1984), 297–306.
V. King: A lower bound for the recognition of digraph properties, Combinatorica10(1) (1990), 53–59.
D. J. Kleitman and D. J. Kwiatkowski: Further results on the Aanderaa-Rosenberg conjecture, J. Combinatorial Theory, Ser. B28(1) (1980), 85–95.
W. S. Massey: A Basic Course in Algebraic Topology, Number 127 in Graduate Texts in Mathematics, Springer, 1991.
R. Oliver: Fixed-point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50(1) (1975), 155–177.
R. L. Rivest and J. Vuillemin: On recognizing graph properties from adjacency matrices, Theor. Comput. Sci. 3(3) (1976/77), 371–384.
A. L. Rosenberg: On the time required to recognize properties of graphs: a problem; SIGACT News5(4) (1973), 15–16.
P. A. Smith: Fixed point theorems for periodic transformations, Amer. J. Math.63 (1941), 1–8.
E. H. Spanier: Algebraic topology, McGraw-Hill series in higher mathematics, McGraw-Hill, New York [u.a.], 1966.
E. Triesch: Some results on elusive graph properties, SIAM J. Comput. 23(2) (1994), 247–254.
E. Triesch: On the recognition complexity of some graph properties, Combinatorica16(2) (1996), 259–268.
A. C. Yao: Monotone bipartite graph properties are evasive, SIAM J. Comput. 17(3) (1988), 517–520.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Korneffel, T., Triesch, E. An asymptotic bound for the complexity of monotone graph properties. Combinatorica 30, 735–743 (2010). https://doi.org/10.1007/s00493-010-2485-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-010-2485-3