Recover constant‑factor welfare in the partially patient and multi‑dimensional extensions

Establish whether there exist online scheduling algorithms that guarantee a constant‑factor approximation to the optimal social welfare under the paper’s average block size framework in (i) the partially patient setting (where values decrease either by a discount factor or after a finite patience window), and (ii) the multi‑dimensional fees setting with multiple resource constraints.

Background

The paper presents impossibility results showing that, in both a partially patient model (mixing patient and partially patient or impatient users) and a multi‑dimensional resource model, any online algorithm must lose at least a fixed constant fraction of welfare.

Despite these lower bounds, it remains open whether one can design algorithms that still achieve a positive constant‑factor approximation (albeit strictly less than one) to the optimal welfare in these extended models.