A Purely Democratic Characterization of W[1]

We give a novel characterization of W[1], the most important fixed-parameter intractability class in the W-hierarchy, using Boolean circuits that consist solely of majority gates. Such gates have a Boolean value of 1 if and only if more than half of their inputs have value 1. Using majority circuits, we define an analog of the W-hierarchy which we call the \(\widetilde{\mathrm{W}}\)-hierarchy, and show that \(\mathrm{W}[1] = \widetilde{\mathrm{W}}[1]\) and \(\mathrm{W}[t] \subseteq \widetilde{\mathrm{W}}[t]\) for all t. This gives the first characterization of W[1] based on the weighted satisfiability problem for monotone Boolean circuits rather than antimonotone. Our results are part of a wider program aimed at exploring the robustness of the notion of weft, showing that it remains a key parameter governing the combinatorial nondeterministic computing strength of circuits, no matter what type of gates these circuits have.