Abstract.
We improve upon indirect diagonalization arguments for lower bounds on explicit problems within the polynomial hierarchy. Our contributions are summarized as follows.
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1.
We present a technique that uniformly improves upon most known nonlinear time lower bounds for nondeterminism and alternating computation, on both subpolynomial (n o(1)) space RAMs and sequential one-tape machines with random access to the input. We obtain improved lower bounds for Boolean satisfiability (SAT), as well as all NP-complete problems that have efficient reductions from SAT, and ∑ k -SAT, for constant k ≥ 2. For example, SAT cannot be solved by random access machines using \(n^{\sqrt{3}}\) time and subpolynomial space.
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2.
We show how indirect diagonalization leads to time-space lower bounds for computation with bounded nondeterminism. For both the random access and multitape Turing machine models, we prove that for all k ≥ 1, there is a constant c k > 1 such that linear time with n 1/k nondeterministic bits is not contained in deterministic \(n^{{c}_{k}}\) time with subpolynomial space. This is used to prove that satisfiability of Boolean circuits with n inputs and n k size cannot be solved by deterministic multitape Turing machines running in \({n^{{k \cdot {c}}_{k}}}\) time and subpolynomial space.
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Manuscript received 19 September 2005
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Williams, R. Inductive Time-Space Lower Bounds for Sat and Related Problems. comput. complex. 15, 433–470 (2006). https://doi.org/10.1007/s00037-007-0221-1
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DOI: https://doi.org/10.1007/s00037-007-0221-1
Keywords.
- Time-space tradeoffs
- lower bounds
- polynomial-time hierarchy
- satisfiability
- diagonalization
- bounded nondeterminism