Inductive Time-Space Lower Bounds for Sat and Related Problems

• Authors: Ryan Williams
• Venue: Computational Complexity
• Year: 2006

We improve upon indirect diagonalization arguments for lower bounds on explicit problems within the polynomial hierarchy. Our contributions are summarized as follows.1. We present a technique that uniformly improves upon most known nonlinear time lower bounds for nondeterminism and alternating computation, on both subpolynomial (no(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 time and subpolynomial space.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 ck 1 such that linear time with n1/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 nk size cannot be solved by deterministic multitape Turing machines running in $${n^{{k \cdot {c}}_{k}}}$$ time and subpolynomial space.