Time-space tradeoffs for counting NP solutions modulo integers

Computational Complexity

Volumn 17, Issue 2, 2008, Pages 179-219

  Williams, R Ryan ^a  

^a Carnegie Mellon University   (United States)

Author keywords

Diagonalization; Lower bounds; Modular counting; Reversible computation; Satisfiability; Time space tradeoffs

Indexed keywords

EID: 44749086443     PISSN: 10163328     EISSN: 14208954     Source Type: Journal    
DOI: 10.1007/s00037-008-0248-y     Document Type: Conference Paper

References (40)

1. E. Allender and V. Gore. A uniform circuit lower bound for the permanent. SIAM Journal on Computing 23:1026-1049, 1994.
2. E. Allender. The permanent requires large uniform threshold circuits. Chicago Journal of Theoretical Computer Science, article 7, 1999.
3. E. Allender, M. Koucky, D. Ronneburger, S. Roy, and V. Vinay. Time-space tradeoffs in the counting hierarchy. In Proceedings of IEEE Conference on Computational Complexity (CCC), 295-302, 2001.
4. D. Barrington, H. Straubing, and D. Thérien. Non-uniform automata over groups. Information and Computation 89(2):109-132, 1990.
5. R. Beigel, H. Buhrman, and L. Fortnow. NP might not be as easy as detecting unique solutions. In Proceedings of ACM Symposium on Theory of Computing (STOC), 203-208, 1998.
6. R. Beigel and J. Gill. Counting classes: thresholds, parity, mods, and fewness. Theoretical Computer Science 103(1):3-23, 1992.
7. C. H. Bennett. Time-space tradeoffs for reversible computation. SIAM Journal on Computing 18:766-776, 1989.
8. J.-Y. Cai and L. A. Hemachandra. On the power of parity polynomial time. Theory of Computing Systems 23(1):95-106, 1990.
9. J.-Y. Cai and V. Choudhary. Some results on matchgates and holographic algorithms. In Proceedings of ICALP Vol. 1, Springer-Verlag LNCS, 703-714, 2006.
10. J.-Y. Cai and P. Lu. Holographic algorithms: from art to science. In Proceedings of ACM Symposium on Theory of Computing (STOC), 401-410, 2007.
11. J.-Y. Cai and P. Lu. Bases collapse in holographic algorithms. In Proceedings of IEEE Conference on Computational Complexity (CCC), 292-304, 2007.
12. 1 computation. Journal of Computer and System Sciences 57(2):200-212, 1998.
13. A. Chattopadhyay, N. Goyal, P. Pudlak, and D. Thérien. Lower bounds for circuits with MODm gates. In Proceedings of IEEE Symposium on Foundations of Computer Science (FOCS), 2006.
14. S. Diehl and D. Van Melkebeek. Time-space lower bounds for the polynomial-time hierarchy on randomized machines. SIAM Journal on Computing 36:563-594, 2006.
15. L. Fortnow. Time-space tradeoffs for satisfiability. J. Comput. Syst. Sci. 60(2):337-353, 2000.
16. L. Fortnow, R. Lipton, D. Van Melkebeek, and A. Viglas. Time-space lower bounds for satisfiability. Journal of the ACM 52(6):835-865, 2005.
17. L. Fortnow and D. Van Melkebeek. Time-space tradeoffs for nondeterministic computation. In Proceedings of IEEE Conference on Computational Complexity (CCC), 2-13, 2000.
18. V. Grolmusz. A degree-decreasing lemma for (MOD-q - MOD-p) circuits. Discrete Mathematics and Theoretical Computer Science 4(2):247-254, 2001.
19. S. Gupta. Isolating an odd number of elements and applications in complexity theory. Theory of Computing Systems 31:27-40, 1998.
20. R. Kannan. Towards separating nondeterminism from determinism. Mathematical Systems Theory 17(1):29-45, 1984.
21. A. Klivans and D. Van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM Journal on Computing 31:1501-1526, 2002.
22. R. Y. Levine and A. T. Sherman. A note on Bennett's time-space tradeoff for reversible computation. SIAM Journal on Computing 19(4):673-677, 1990.
23. R. J. Lipton and A. Viglas. On the complexity of SAT. In Proceedings of IEEE Symposium on Foundations of Computer Science (FOCS), 459-464, 1999.
24. W. Maass and A. Schorr. Speed-up of Turing machines with one work tape and a two-way input tape. SIAM Journal on Computing 16(1):195-202, 1987.
25. D. Van Melkebeek. Time-space lower bounds for NP-complete problems. In G. Paun, G. Rozenberg, and A. Salomaa (eds.), Current Trends in Theor. Comp. Sci. 265-291, World Scientific, 2004.
26. D. Van Melkebeek and R. Raz. A time lower bound for satisfiability. Theoretical Computer Science 348(2-3):311-320, 2005.
27. D. Van Melkebeek. A survey of lower bounds for satisfiability and related problems. Foundations and Trends in Theoretical Computer Science 2(3):197-303, 2007.
28. A. V. Naik, K. W. Regan, D. Sivakumar. On quasilinear-time complexity theory. Theoretical Computer Science 148:325-349, 1995.
29. V. Nepomnjascii. Rudimentary predicates and Turing calculations. Soviet Math. Doklady 11:1462-1465, 1970.
30. C. H. Papadimitriou and S. Zachos. Two remarks on the power of counting. Theoretical Computer Science 1983, Springer LNCS 145:269-276, 1983.
31. R. Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Proceedings of ACM Symposium on Theory of Computing (STOC), 77-82, 1987.
32. S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. SIAM Journal on Computing 21(2):316-328, 1992.
33. I. Tourlakis. Time-space tradeoffs for SAT on nonuniform machines. Journal of Computer and System Sciences 63(2):268-287, 2001.
34. L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science 47(1):85-93, 1986.
35. L. Valiant. Holographic algorithms. In Proceedings of IEEE Symposium on Foundations of Computer Science (FOCS), 306-315, 2004.
36. L. Valiant. Accidental algorithms. In Proceedings of IEEE Symposium on Foundations of Computer Science (FOCS), 509-517, 2006.
37. R. Williams. Inductive time-space lower bounds for SAT and related problems. Computational Complexity 15:433-470, 2006.
38. R. Williams. Algorithms and resource requirements for fundamental problems. Ph.D. Thesis, Carnegie Mellon University, CMU-CS-07-147, August 2007.
39. R. Williams. Automated proofs of time lower bounds. Manuscript, 2007.
40. M. Xia, P. Zhang, and W. Zhao. Computational complexity of counting problems on 3-regular planar graphs. Theoretical Computer Science 384(1):111-125, 2007.