A note on width-parameterized SAT: An exact machine-model characterization

We characterize the complexity of SAT instances with path-decompositions of width w(n). Although pathwidth is the most restrictive among the studied width-parameterizations of SAT, the most time-efficient algorithms known for such SAT instances run in time 2, even when the path-decomposition is given in the input. We wish to better understand the decision complexity of SAT instances of width ω(log n). We provide an exact correspondence between SATpw[w(n)], the problem of SAT instances with given path decomposition of width w(n), and NL[r(n)], the class of problems decided by logspace Turing Machines with at most r(n) passes over the nondeterministic tape. In particular, we show that SATpw[w(n)] is hard for NL[ log n ] under log-space reductions. When NL[ w(n) log n ] is closed under logspace reductions, which is the case for the most interesting w(n)’s, we show that SATpw[w(n)] is also complete.