Exact complexity of the parameterized Ckt-Condorcet problem for k ≥ 3

Determine the exact complexity classification of the decision problem Ckt-Condorcet[k] for any fixed integer k ≥ 3, where the input consists of k Boolean circuits each mapping n-bit strings to n-bit outputs and the question asks whether there exists an n-bit string that wins a pairwise majority comparison against every other n-bit string. Specifically, establish whether Ckt-Condorcet[k] is complete for a known class (such as Σ2^P or PCW), or identify its precise position within the polynomial hierarchy or the unambiguous subclasses considered in the paper.

Background

The paper defines the class PCW (Polynomial Condorcet Winner) via the Ckt-Condorcet problem, where candidates are exponentially many strings and pairwise comparisons are computed by a polynomial-size set of circuits, each producing a numeric score; an n-bit string is a Condorcet winner if it beats every other string by majority vote across the circuits.

A parameterized variant Ckt-Condorcet[k] fixes the number of circuits to k. The authors show that for k=1 and k=2 the problem is Σ2^P-complete, while for k≥3 the problem is known to be Σ2^P-hard and in PCW, but its exact complexity status remains unresolved.