Upper Bounds for Vertex Cover Further Improved
The problem instance of Vertex Cover consists of an undirected graph G = (V, E) and a positive integer k, the question is whether there exists a subset C ⊂-V of vertices such that each edge in E has at least one of its endpoints in C with |C|≤ k. We improve two recent worst case upper bounds for Vertex Cover. First, Balasubramanian et al. showed that Vertex Cover can be solved in time O(kn + 1.32472k k 2), where n is the number of vertices in G. Afterwards, Downey et al. improved this to O(kn + 1.31951k k 2). Bringing the exponential base significantly below 1.3, we present the new upper bound O(kn + 1.29175k k 2).
KeywordsSearch Tree Vertex Cover Optimal Cover Marked Vertex Search Tree Algorithm
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- S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and hardness of approximation problems. In Proceedings of the 33d IEEE Conference on Foundations of Computer Science, pages 14–23, 1992.Google Scholar
- P. Crescenzi and V. Kann. A compendium of NP optimization problems. Available at http://www.nada.kth.se/theory/problemlist.html, April 1997.
- R. G. Downey and M. R. Fellows. Parameterized Complexity. Springer-Verlag, 1998.Google Scholar
- R. G. Downey, M. R. Fellows, and U. Stege. Parameterized complexity: A framework for systematically confronting computational intractability. In F. Roberts, J. Kratochvíl, and J. Nesetril, editors, The Future of Discrete Mathematics: Proceedings of the First DIMATIA Symposium, June 1997, AMS-DIMACS Proceedings Series. AMS, 1998. To appear. Available through http://www.inf.ethz.ch/personal/stege.
- R. C. Evans. Testing repairable RAMs and mostly good memories. In Proceedings of the IEEE Int. Test Conf., pages 49–55, 1981.Google Scholar
- M. Hallett, G. Gonnet, and U. Stege. Vertex cover revisited: A hybrid algorithm of theory and heuristic. Manuscript, 1998.Google Scholar
- J. Håstad. Some optimal inapproximability results. In Proceedings of the 29th ACM Symposium on Theory of Computing, pages 1–10, 1997.Google Scholar
- O. Kullmann and H. Luckhardt. Deciding propositional tautologies: Algorithms and their complexity. 1997. Submitted to Information and Computation.Google Scholar
- R. Niedermeier. Some prospects for efficent fixed parameter algorithms (invited paper). In B. Rovan, editor, Proceedings of the 25th Conference on Current Trends in Theory and Practice of Informatics (SOFSEM), number 1521 in Lecture Notes in Computer Science, pages 168–185. Springer-Verlag, 1998.Google Scholar
- R. Niedermeier and P. Rossmanith. Upper bounds for vertex cover further improved. Technical Report KAM-DIMATIA Series 98-411, Faculty of Mathematics and Physics, Charles University, Prague, November 1998.Google Scholar
- C. H. Papadimitriou. Computational Complexity. Addison-Wesley, 1994.Google Scholar