A Quadratic Lower Bound for the Shield Number of the Stable Marriage Problem: Rearrangement, Extremal Construction, and Biclique Realizability

Abstract

We study the Shield number of the stable marriage problem, defined as the minimum, over all preference profiles of size n, of the maximum number of blocking pairs attainable by a complete matching. We establish the universal quadratic lower bound $σ(n) \ge \lceil n(n-2)/6 \rceil$ using a rearrangement inequality and show that this bound is the strongest consequence obtainable from uniform first-moment arguments. We further prove a sharp min-rearrangement inequality and construct an explicit cyclic Hollow-Shell instance attaining $\lfloor (n-1)^2/4 \rfloor$ blocking pairs. We also establish sufficient Hall-type conditions for biclique realizability, prove structural limitations of uniform averaging and two-parameter linear assignment objectives, and provide complete proofs for all unconditional results. Computer-assisted verification is used only for explicitly identified computational statements and is clearly distinguished from mathematical proofs.