## 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.

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Citation Context ...) > n such that MODpTIME[t] � MODqTIME[o(t)]. The hypothesis looks very reasonable in light of current knowledge, such as the circuit lower bounds for computing MODp with OR, AND, NOT, and MODq gates =-=[Smo87]-=-. It seems extremely counterintuitive that counting solutions modulo one prime would somehow always be faster, if one could count modulo a different prime. If the hypothesis is true, then indeed it fo... |

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Citation Context ...for polynomial time, for every composite m that is not a prime power. 1.2. Other Related Work. Modulo Counting Classes. The class MOD2P (also written as ⊕P) was introduced by Papadimitriou and Zachos =-=[PZ83]-=-, and the class MODkP for arbitrary k was introduced by Cai and Hemachandra [CH90]. Beigel and Gill [BG92] showed several closure properties hold for these classes—for example, for all primes p, the c... |

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Citation Context ...tion also serves as a reduction from an arbitrary Lk ∈ MODkTIME[n] to MODk-Sat. ✷ 3 Related Work Modulo Counting Classes. The class MODkP was introduced by Cai and Hemachandra [CH89]. Beigel and Gill =-=[BG92]-=- showed several closure properties hold for these classes– for example, for all primes p, the class MODpP is closed under union, intersection, and complement. Thus it is strongly believed that MODpP �... |

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Citation Context ...by k, so the above reduction also serves as a reduction from an arbitrary Lk ∈ MODkTIME[n] to MODk-Sat. ✷ 3 Related Work Modulo Counting Classes. The class MODkP was introduced by Cai and Hemachandra =-=[CH89]-=-. Beigel and Gill [BG92] showed several closure properties hold for these classes– for example, for all primes p, the class MODpP is closed under union, intersection, and complement. Thus it is strong... |

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Citation Context ...ing Hierarchy. Several interesting lower bounds on counting problems have been discovered in the past. Allender and Gore [AG94] proved that uniform ACC 0 is properly contained in PP. Caussinus et al. =-=[CMTV98]-=- showed that uniform ACC 0 is properly contained in MODPH (a counting version of the polynomial hierarchy). Allender [All98] showed that the Permanent is not in TC 0 , thus TC 0 �= PP. Allender et al.... |

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Citation Context ...appears that we might be able to extend such results without much trouble, given that NP can be reduced to MODpP (for all primes p) by randomized reductions, via the well-known Valiant-Vazirani lemma =-=[VV86]-=-. A quasilinear time version of Valiant and Vazirani’s reduction has been found by Naik, Regan, and Sivakumar [NRS95]. Moreover, Toda and Ogihara [TO92] showed that the entire polynomial hierarchy red... |

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Citation Context ... S(n)] is contained in MODqMODpTIME[(T (n)S(n)) 1/2+ε ], for all ε > 0. 2sUsing Theorem 1.1 and some elementary number theory, much of the research on time-space tradeoff lower bounds for Sat on RAMs =-=[FLvMV05]-=- can be directly transferred over to MODp-Sat. Independently of the MODp-Sat transfer arguments, we also add a new argument to this line of work, culminating in a new collection of time-space lower bo... |

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Citation Context ...d complement. Thus it is strongly believed that MODpP �= NP for all primes p. In contrast to the Valiant-Vazirani lemma and Toda’s theorem, which show the power of MODkP, Beigel, Buhrman, and Fortnow =-=[BBF98]-=- demonstrated a very interesting oracle collapse/separation for MOD classes. A consequence of their oracle construction is that for all distinct primes p and q, there is an oracle A such that P A = MO... |

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Citation Context ...there is a k ≥ 1 such that for all j �= k, NTIME[n j ] � DTISP[n j , o(n j/k )], thus establishing a time-space tradeoff for nondeterminism. This line of study was revived in the late 90’s by Fortnow =-=[For97]-=-, who proved that Sat /∈ DTIME[n 1+o(1) ] ∩ NL. Fortnow-Lipton-ViglasVan Melkebeek [FLvMV05] sharpened the tools and arguments, showing that Sat is not in n φ−ε time and n o(1) space for all ε > 0, wh... |

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Citation Context ...ork shows the power of MODpq-Sat, as the known proofs of lower bounds for Sat can be extended to it. Time-Space Tradeoffs for Nondeterminism. Using a variant on Nepomnjascii’s theorem [Nep70], Kannan =-=[Kan84]-=- showed that there is a k ≥ 1 such that for all j �= k, NTIME[n j ] � DTISP[n j , o(n j/k )], thus establishing a time-space tradeoff for nondeterminism. This line of study was revived in the late 90’... |

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Citation Context ...showed that uniform ACC 0 is properly contained in MODPH (a counting version of the polynomial hierarchy). Allender [All98] showed that the Permanent is not in TC 0 , thus TC 0 �= PP. Allender et al. =-=[AKRRV01]-=- proved time-space tradeoffs for MAJ-MAJ-SAT (MAJ-MAJ-SAT is a complete problem in the second level of the counting hierarchy, a generalization of MAJ-SAT), showing that it is not contained in unbound... |

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Citation Context ...ulas with a finite number of quantifier blocks have also been discovered, with larger exponents [FLVV05, Wil06, Wil07a]. All the above results use a form of argument known as indirect diagonalization =-=[vM04]-=-, which proceeds by showing that the negation of the lower bound one wants to prove implies a contradiction with a known time hierarchy theorem. Our goal here is to develop methods for extending the r... |

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Citation Context ...teger k ≥ 1, there is a reversible machine M ′ that runs in T 1+1/k (n) time and 2 k S(n) log 2 T (n) space, such that L(M) = L(M ′ ). Along with Bennett [Ben89], see also Buhrman, Tromp, and Vitanyi =-=[BTV01]-=- for an exposition of the proof. We require two specific properties of the reversible M ′ obtained in Theorem 5.1. Remark 1 The reversible M ′ of Theorem 5.1 has the following additional properties: 1... |

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Citation Context ... time and n o(1) space for all ε > 0, where φ is the golden ratio. Earlier work of ours [Wil05] improved the time lower bound to greater than n √ 3 . Building on our argument, Diehl and Van Melkebeek =-=[DvM06]-=- gained a slight improvement to n 1.759 . Lower Bounds for the Counting Hierarchy. Several interesting lower bounds on counting problems have been discovered in the past. Allender and Gore [AG94] prov... |

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Citation Context ...e formula checks that the temporally ordered sequence is a permutation of the k spatially ordered sequences (this is the most technical step, which can be achieved by using efficient sorting networks =-=[vM07]-=-), and another formula uses the variables of the spatially ordered sequence to verify that (1) when a tape cell is first read, it contains the appropriate value, and (2) every symbol read in a tape ce... |

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Citation Context ...Fortnow-Lipton-ViglasVan Melkebeek [FLvMV05] sharpened the tools and arguments, showing that Sat is not in n φ−ε time and n o(1) space for all ε > 0, where φ is the golden ratio. Earlier work of ours =-=[Wil05]-=- improved the time lower bound to greater than n √ 3 . Building on our argument, Diehl and Van Melkebeek [DvM06] gained a slight improvement to n 1.759 . Lower Bounds for the Counting Hierarchy. Sever... |

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Citation Context ...d that the entire polynomial hierarchy reduces to MODpP using two-sided randomized reductions (but for ΣkP where k ≥ 2, the best reduction we know of from ΣkP to MODpP takes Θ(n k+1 ) time, cf. Gupta =-=[Gup98]-=-). However, despite their time-efficiency, the inherent randomness of these reductions is a major difficulty in applying them to obtain time-space lower bounds, since we do not know how to remove the ... |

5 |
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Citation Context ...mple is the problem of counting satisfying assignments to a planar read-twice monotone 3-CNF formula. In spite of the numerous adjectives restricting the problem, this counting problem is #P-complete =-=[XZZ07]-=-; however, counting the number of satisfying assignments to such a formula modulo 7 turns out to be in P [Val06, CL07a]. Such results challenge our basic intuitions about the complexity of counting. I... |

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1 | A Degree-Decreasing Lemma for (MOD-q - MOD-p - Grolmusz |

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Citation Context ...or all primes p) by randomized reductions, via the well-known ValiantVazirani lemma [VV86]. A quasilinear time version of Valiant and Vazirani’s reduction has been found by Naik, Regan, and Sivakumar =-=[NRS95]-=-. Moreover, Toda and Ogihara [TO92] showed that the entire polynomial hierarchy reduces to MODpP using two-sided randomized reductions (but for ΣkP where k ≥ 2, the best reduction we know of from ΣkP ... |