## Time-Space Tradeoffs for Counting NP Solutions Modulo Integers (2007)

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Venue: | In Proceedings of the 22nd IEEE Conference on Computational Complexity |

Citations: | 11 - 5 self |

### BibTeX

@INPROCEEDINGS{Williams07time-spacetradeoffs,

author = {R. Ryan Williams},

title = {Time-Space Tradeoffs for Counting NP Solutions Modulo Integers},

booktitle = {In Proceedings of the 22nd IEEE Conference on Computational Complexity},

year = {2007},

pages = {70--82},

publisher = {IEEE}

}

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### Abstract

We prove the first time-space tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon known time-space tradeoffs for Sat. Let m> 0 be an integer, and define MODm-Sat to be the problem of determining if a given Boolean formula has exactly km satisfying assignments, for some integer k. We show for all primes p except for possibly one of them, and for all c < 2cos(π/7) ≈ 1.801, there is a d> 0 such that MODp-Sat is not solvable in n c time and n d space by general algorithms. That is, there is at most one prime p that does not satisfy the tradeoff. We prove that the same limitation holds for Sat and MOD6-Sat, as well as MODm-Sat for any composite m that is not a prime power. Our main tool is a general method for rapidly simulating deterministic computations with restricted space, by counting the number of solutions to NP predicates modulo integers. The simulation converts an ordinary algorithm into a “canonical ” one that consumes roughly the same amount of time and space, yet canonical algorithms have nice properties suitable for counting.