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].