Are quadratic essential coalitions sufficient to compute the nucleolus in many-to-one assignment games?

Determine whether, in many-to-one assignment games, the quadratic-size set of essential coalitions used to describe the core via workers’ salary constraints (with respect to an optimal matching) is sufficient to compute the nucleolus by lexicographic minimization of coalition excesses; equivalently, ascertain whether restricting the nucleolus computation to only those essential coalitions yields the true nucleolus without considering the full exponential collection of coalitions.

Background

The nucleolus of a cooperative game is defined by lexicographically minimizing the ordered excesses over all coalitions, which in general requires handling an exponential number of constraints. In many-to-one assignment games, the authors provide a quadratic-size description of the core in terms of workers’ payoffs relative to an optimal matching (Proposition 3), dramatically reducing the constraints needed to characterize the core.

The open question asks whether this reduced, quadratic set of essential coalitions suffices for determining the nucleolus, i.e., whether the lexicographic minimization can be restricted to these coalitions without altering the nucleolus outcome. Establishing this would yield a significant computational simplification for nucleolus computation in many-to-one assignment markets.