Strict separation between edge-weighted and vertex-weighted factor-revealing values

Determine whether the edge-weighted factor-revealing value is strictly smaller than the corresponding value obtained when complementary monotone gain-sharing rules and the vertex-weighted cutoff constraint are allowed; equivalently, prove or refute the conjectured strictness of the first inequality in the factor-revealing hierarchy.

Background

The paper defines a hierarchy of factor-revealing values for edge-weighted, vertex-weighted, and unweighted matching analyses. The edge-weighted value optimizes over the score-balanced gain-sharing subclass and permits arbitrary cutoff curves, whereas the vertex-weighted value optimizes over the larger class of complementary monotone gain-sharing rules and restricts one cutoff curve to be nondecreasing.

Numerical computations indicate that imposing the additional vertex-weighted cutoff monotonicity constraint does not improve the optimum when the larger gain-sharing class is used. However, the authors observe no structural reason for an optimal gain-sharing rule in that larger class to be score-balanced, which motivates the conjecture that the edge-weighted value is strictly smaller than the vertex-weighted value. Resolving this would identify whether the separation arises from the gain-sharing class rather than from cutoff geometry.