Parameterized Circuit Complexity and the W Hierarchy

Abstract A parameterized problem 〈 L , k 〉 belongs to W [ t ] if there exists k ′ computed from k such that 〈 L , k 〉 reduces to the weight-k′ satisfiability problem for weft- t circuits. We relate the fundamental question of whether the W [ t ] hierarchy is proper to parameterized problems for constant-depth circuits. We define classes G [ t ] as the analogues of AC 0 depth- t for parameterized problems, and N [ t ] by weight- k ′ existential quantification on G [ t ], by analogy with NP = ∃ · P . We prove that for each t , W [ t ] equals the closure under fixed-parameter reductions of N [ t ]. Then we prove, using Sipser's results on the AC 0 depth- t hierarchy, that both the G [ t ] and the N [ t ] hierarchies are proper. If this separation holds up under parameterized reductions, then the W [ t ] hierarchy is proper. We also investigate the hierarchy H [ t ] defined by alternating quantification over G [ t ]. By trading weft for quantifiers we show that H [ t ] coincides with H [1]. We also consider the complexity of unique solutions, and show a randomized reduction from W [ t ] to Unique W [ t ].