NP-hardness of the Balanced Biclique-Or-No-Partial-Multi-Cover promise problem

Determine whether the Balanced Biclique-Or-No-Partial-Multi-Cover promise problem is NP-hard.

Background

The paper introduces the promise problem Balanced Biclique-Or-No-Partial-Multi-Cover (Bb-Pmc), which distinguishes between a bipartite graph containing a balanced biclique of a specified size and a graph containing no partial cover of that size. A polynomial-time algorithm for EJR on committee elections with utilities bounded by twice the number of voters would place Bb-Pmc in P. Consequently, proving Bb-Pmc NP-hard would imply that no such polynomial-time EJR rule exists unless P equals NP; the required NP-hardness is left unresolved.