Combining one-half PROP1 with the optimal normalized envy bound

Establish whether 1/2-PROP1 can be combined with an O(\sqrt{T\log n/n}) normalized maximum additive envy guarantee for online allocation with exact maximum item value predictions and a known horizon.

Background

With exact MIV predictions and a known horizon, the paper proves that every fixed PROP1 factor β<1/2 can be combined with O(\sqrt{T\log n/n}) normalized maximum additive envy when T is sufficiently large relative to n\log n.

The proof requires β<1/2, and the authors explicitly leave unresolved whether the boundary guarantee 1/2-PROP1 can be achieved simultaneously with the same envy rate.