Abstract
We provide new non-approximability results for the restrictions of the Min Vertex Cover problem to bounded-degree, sparse and dense graphs. We show that, for a sufficiently large B, the recent 16/15 lower bound proved by Bellare et al. [3] extends with negligible loss to graphs with bounded degree B. Then, we consider sparse graphs with no dense components (i.e. everywhere sparse graphs), and we show a similar result but with a better trade-off between non-approximability and sparsity. Finally we observe that the Min Vertex Cover problem remains APX-complete when restricted to dense graph and thus recent techniques developed by Arora et al. [1] for several Max SNP problems restricted to “dense” instances cannot be applied.
On leave from the Centre Universitaire d'Informatique of the University of Geneve. Partially supported by the Swiss National Science Foundation, grant 21-43309.95.
Part of this research has been done while this author was visiting the Centre Universitaire d'Informatique of the University of Geneve partially supported by the HCM project SCOOP of the European Union.
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References
S. Arora, D. Karger, and M. Karpinski. Polynomial time approximation schemes for dense instances of graph problems. In Proceedings of the 27th ACM Symposium on Theory of Computing, pages 284–293, 1995.
S. Arora and S. Safra. Probabilistic checking of proofs; a new characterization of NP. In Proceedings of 33rd Symposium on Foundations of Computer Science, pages 2–13, 1992.
M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCP's and non-approximability — towards tight results (3rd version). Technical Report TR95-24, Electronic Colloquium on Computational Complexity, 1995. Extended abstract in Proc. of FOCS'95.
P. Berman and T. Fujito. On the approximation properties of the independent set problem in degree 3 graphs. In Proceedings of the 4th Workshop on Algorithms and Data Structures, pages 449–460. LNCS 955, Springer-Verlag, 1995.
M. Bern and P. Plassmann. The steiner tree problem with edge lengths 1 and 2. Information Processing Letters, 32:171–176, 1989.
A.E.F. Clementi and L. Trevisan. Improved non-approximability results for vertex cover with density constraints. Technical Report TR 96-16, Electronic Colloquium on Computational Complexity, 1996.
P. Crescenzi and V. Kann. A compendium of NP optimization problems. Technical Report TR SI-95/02, Università di Roma “La Sapienza”, Dipartimento di Scienze dell'Informazione, 1995. Updated on-line version is available at the URL http://www.nada.kth.se/∼viggo/problemlist/compendium.html.
P. Crescenzi and A. Panconesi. Completeness in approximation classes. Information and Computation, 93:241–262, 1991. Preliminary version in Proc. of FCT'89.
U. Feige. A threshold of ln n for approximating set cover. In Proceedings of the 28th ACM Symposium on Theory of Computing, 1996. To appear.
U. Feige, S. Goldwasser, L. Lovasz, S. Safra, and M. Szegedy. Approximating clique is almost NP-complete. In Proceedings of 32nd Symposium on Foundations of Computer Science, pages 2–12, 1991.
M. R. Garey, D. S. Johnson, and L. Stockmeyer. Some simplified NP-complete graph problems. Theoretical Computer Science, 1:237–267, 1976.
M.R. Garey and D.S. Johnson. Computers and Intractability: a Guide to the Theory of NP-Completeness. Freeman, 1979.
F. Gavril. Manuscript cited in [12], 1974.
J. Håstad. Testing of the long code and hardness for clique. In Proceedings of the 28th ACM Symposium on Theory of Computing, 1996. To appear.
R.M. Karp. Reducibility among combinatorial problems. In R.E. Miller and J.W. Thatcher, editors, Complexity of Computer Computations, pages 85–103. Plenum Press, 1972.
S. Khanna, R. Motwani, M. Sudan, and U. Vazirani. On syntactic versus computational views of approximability. In Proceedings of the 35th IEEE Symposium on Foundations of Computer Science, pages 819–830, 1994.
A. Lubotzky, R. Phillips, and P. Sarnak. Ramanujan graphs. Combinatorica, 8(3):261–277, 1988.
B. Monien and E. Speckenmeyer. Some further approximation algorithms for the vertex cover problem. In Proceedings of CAAP83, pages 341–349. LNCS 159, Springer Verlag, 1983.
C. H. Papadimitriou and M. Yannakakis. Optimization, approximation, and complexity classes. Journal of Computer and System Sciences, 43:425–440, 1991. Preliminary version in Proc. of STOC'88.
C.H. Papadimitriou. Computational Complexity. Addison-Wesley, 1993.
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Clementi, A.E.F., Trevisan, L. (1996). Improved non-approximability results for vertex cover with density constraints. In: Cai, JY., Wong, C.K. (eds) Computing and Combinatorics. COCOON 1996. Lecture Notes in Computer Science, vol 1090. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61332-3_167
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DOI: https://doi.org/10.1007/3-540-61332-3_167
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