Abstract
The main problem we consider in this paper is the Independent Set problem for bounded degree graphs. It is shown that the problem remains MAX SNP-complete when the maximum degree is bounded by 3. Some related problems are also shown to be MAX SNP-complete at the lowest possible degree bounds. Next we study better poly-time approximation of the problem for degree 3 graphs, and improve the previously best ratio, 5/4, to arbitrarily close to 6/5. This result also provides improved poly-time approximation ratios, B+3/5+ε, for odd degree B.
This work was partially supported by NSF Grant CCR-9114545
Part of this work was done while the author was at Dept. of CSE, Penn State.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
S. Arora, C. Lund, R. Motwani, M. Sudan and M. Szegedy. Proof verification and intractability of approximation problems. In 33rd IEEE Symp. on Foundations of Computer Science, 1992.
B.S. Baker. Approximation algorithms for NP-complete problems on planar graphs. Journal of the Association for Computing Machinery, 41:153–180, 1994.
R. Bar-Yehuda, D. Geiger, J. Naor and R. M. Roth. Approximation algorithms for the vertex feedback set problem with applications to constraint satisfaction and bayesian inference. In Proc. of the 5th Annual ACM-SIAM Symp. on Discrete Algorithms, pages 344–354, 1994.
P. Berman and M. Fürer. Approximating maximum independent set in bounded degree graphs. In Proc. of the 5th Annual ACM-SIAM Symp. on Discrete Algorithms, 1994.
R.L. Brooks. On coloring the nodes of a network. In Proc. Cambridge Philos. Soc., volume 37, pages 194–197, 1941.
M.R. Garey, D.S. Johnson and L. Stockmeyer. Some simplified NP-complete graph problems. Theoretical Computer Science, 1:237–267, 1976.
M.M. Halldórsson and J. Radhakrishnan. Greed is good: Approximating independent sets in sparse and bounded-degree graphs. In Proc. of the 26th Annual ACM Symp. on Theory of Computing, 1994.
M.M. Halldórsson and J. Radhakrishnan. Improved approximations of independent sets in bounded-degree graphs. In SWAT 94, 6th Scandinavian Workshop on Algorithm Theory, 1994.
D.S. Hochbaum. Efficient bounds for the stable set, vertex cover and set packing problems. Discrete Applied Mathematics, 6:243–254, 1983.
V. Kann. Maximum bounded 3-dimensional matching is MAX SNP-complete. Information Processing Letters, 37:27–35, 1991.
S. Khanna, R. Motwani, M. Sudan and U. Vazirani. On syntactic versus computation views of approximability. In 35th IEEE Symp. on Foundations of Computer Science, pages 819–830, 1994.
L. Lovász. Three short proofs in graph theory. Journal of Combinatorial Theory (B), 19:269–271, 1975.
G.L. Nemhauser and L.E. Trotter Jr.. Vertex packings: structural properties and algorithms. Mathematical Programming, 8:232–248, 1975.
C. Papadimitriou and M. Yannakakis. Optimization, approximation and complexity classes. Journal of Computer and System Sciences, 43:425–440, 1991.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Berman, P., Fujito, T. (1995). On approximation properties of the Independent set problem for degree 3 graphs. In: Akl, S.G., Dehne, F., Sack, JR., Santoro, N. (eds) Algorithms and Data Structures. WADS 1995. Lecture Notes in Computer Science, vol 955. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60220-8_84
Download citation
DOI: https://doi.org/10.1007/3-540-60220-8_84
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60220-0
Online ISBN: 978-3-540-44747-4
eBook Packages: Springer Book Archive