# Probabilistic log-space reductions and problems probabilistically hard for p

## Abstract

We present in this paper an interesting kind of reductions and its variations (the probabilistic NC reductions) which allow us to argue about the parallel complexity of problems not easily shown to be complete for P. We show that a problem complete under these reductions cannot be in NC unless P c RNC. Based on these reductions we also show the P-completeness of a probabilistic problem, namely, given a dag for some nodes of which exactly one of the incoming edges fail (all incoming edges to such a node have equal probability to fail) approximate, within an arbitrary given absolute performance ratio the longest distance we can travel, with certainty, along the edges of the dag.

In order to show the above result we show the P-completeness of the problem of approximating, with a given absolute performance ratio, the depth at which the ones arrive in a boolean monotone circuit.

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## References

- "Approximating P-complete problems", by Anderson R. and Mayr E., Tech. Report, Stanford University.Google Scholar
- [Borodin, Von Zur Gathen, Horpcroft, 82] "Fast parallel matrix and CCD computations" by Borodin A., Von Zur Gathen J. and Hopcroft J., computations" by Borodin A., Von Zur Gathen J. and Hopcroft J., Inform. and Control 52, 241–256.Google Scholar
- [Cook, 85] "A Taxonomy of Problems with Fast Parallel Algorithms" by S. Cook, Inform. and Control 64, 2–22, 1985.Google Scholar
- [Dopkin, Lipton, Reiss, 1979] "Linear programming is log-space hard for p" Inform. Process. Lett. 8, 96–97.Google Scholar
- [Garey, Johnson, 79] "Computers and Intractability — A Guide to the Theory of NP-Completeness", by M. Garey and D. Johnson, Free and Co, New York, 1979.Google Scholar
- [Goldschlager, 1977] "The Monotone and Planar circuit value problems and log space complete for p" by L.M. Goldschlager, DIGACT News 9, no. 2, 25–29.Google Scholar
- [Goldschlager, Shaw, Staples, 82] "The maximum flow problem is log space complete for P", by L.M. Goldschlager, R.A. Shaw and J. Staples, Theoret. Comp. Science, 21, pp. 105–111, 1982.Google Scholar
- "A compendium of problems complete for P" by H. Hoover and M.L. Ruzzo, unpublished manuscript, 1984.Google Scholar
- "Languages which Capture Complexity Classes" by N. Immerman, 15th STOC Symposium, pp. 347–354, 1983.Google Scholar
- [Khanchian, 1979] "A polynomial time algorithm for linear programming" Dokl. Akad. Nauk SSSP 244 no. 5, 1093–96 by L.G. Khanchian, transl. in Soviet Math. Dokl. 20, 191–194.Google Scholar
- [Ladner, 75] "The Circuit Value Problem is log-space complete for P" SIGACT News 7, no. 1, 18–20, by R.E. Ladner, 1975.Google Scholar
- "Nondeterministic log-space reductions" by K.-J. Lange in M.P. Chytil and V. Koubek (Eds.): Mathematical Foundations of Computer Science 1984, Lect. Notes Comp. Sci. 176 (1984) 378–388, Springer-Verlag, Berlin-Heidelberg-New York.Google Scholar
- [Lawler, 1976] "Combinatorial Optimization, Networks and Matroids" by E.L. Lawler; Holt, Rinehart and Winston, New York, 1976.Google Scholar
- [Vazirani, Vazirani, 82] "A Natural Encoding Scheme Proved Probabilistic Polynimial Complete" by U. Vazirani and Vijay Vazirani, 23rd FOCS Symposium, 1982, pp. 40–44.Google Scholar
- [Warmuth, 87] "Parallel Approximation Algorithm for One-Dimensional Bin Packing", by M. Warmuth, tech. report, Univ. of California Santa Cruz, 1987.Google Scholar