Relating Partial and Complete Solutions and the Complexity of Computing Smallest Solutions
We prove that computing a single pair of vertices that are mapped onto each other by an isomorphism ϕ between two isomorphic graphs is as hard as computing ϕ itself.Th is result optimally improves upon a result of Gál et al.W e establish a similar, albeit slightly weaker, result about computing complete Hamiltonian cycles of a graph from partial Hamiltonian cycles.W e also show that computing the lexicographically first four-coloring for planar graphs is δ 2 p -hard. This result optimally improves upon a result of Khuller and Vazirani who prove this problem to be NP-hard, and conclude that it is not self-reducible in the sense of Schnorr, assuming P ≠ NP. W e discuss this application to non-self-reducibility and provide a general related result.
Keywordspartial solutions complexity of smallest solutions selfreducibility graph isomorphisms Hamiltonian cycles graph colorability
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- 3.A. Borodin and A. Demers. Some comments on functional self-reducibility and the NP hierarchy. Technical Report TR 76-284, Cornell Department of Computer Science, Ithaca, NY, July 1976.Google Scholar
- 4.A. Gál, S. Halevi, R. Lipton, and E. Petrank. Computing from partial solutions. In Proceedings of the 14th Annual IEEE Conference on Computational Complexity, pages 34–45. IEEE Computer Society Press, May 1999.Google Scholar
- 7.A. Große. Partielle Lösungen NP-vollständiger Probleme. Diploma thesis, Friedrich-Schiller-Universität Jena, Institut für Informatik, Jena, Germany, December 1999. In German.Google Scholar
- 10.D. Joseph and P. Young. Self-reducibility: Effects of internal structure on computational complexity. In A. Selman, editor, Complexity Theory Retrospective, pages 82–107. Springer-Verlag, 1990.Google Scholar
- 14.A. Meyer and M. Paterson. With what frequency are apparently intractable problems difficult? Technical Report MIT/LCS/TM-126, MIT Laboratory for Computer Science, Cambridge, MA, 1979.Google Scholar
- 16.C. Schnorr. Optimal algorithms for self-reducible problems. In S. Michaelson and R. Milner, editors, Proceedings of the 3rd International Colloquium on Automata, Languages, and Programming, pages 322–337, University of Edinburgh, July 1976. Edinburgh University Press.Google Scholar
- 17.C. Schnorr. On self-transformable combinatorial problems, 1979. Presented at IEEE Symposium on Information Theory, Udine, and Symposium über Mathematische Optimierung, Oberwolfach.Google Scholar
- 20.H. Vollmer. On different reducibility notions for function classes. In Proceedings of the 11th Annual Symposium on Theoretical Aspects of Computer Science, pages 449–460. Springer-Verlag Lecture Notes in Computer Science #775, February 1994.Google Scholar