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

Abstract

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 Ω ( w ( n ) ) , 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 SAT pw [ 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 SAT pw [ w ( n ) ] is hard for NL [ w ( n ) 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 SAT pw [ w ( n ) ] is also complete.