Randomized algorithms and the large-inventory threshold

Determine whether randomized algorithms for the online trading problem—where suppliers and customers arrive online, each customer bundle has maximum size d and values lie in [1, v], and the benchmark is an optimal offline fractional solution with supplier valuations augmented by a factor (1+ε)—can achieve a finite competitive ratio without requiring that, for each item type i, the inventory capacity w_i be at least (c/ε)·log(2dv) times the maximum quantity of type i appearing in any bundle, for some constant c>0.

Background

The paper proves logarithmic competitive algorithms for the online trading problem under a large-inventory assumption and with resource augmentation on supplier values. Specifically, for deterministic algorithms the inventory capacity for each item type must scale as Θ((1/ε)·log(2dv)) times the maximum per-bundle quantity of that item to guarantee a finite competitive ratio.

They also show a threshold phenomenon: if the inventory is below a certain multiple of (1/ε)·log(2dv), no deterministic algorithm can have a finite competitive ratio. Whether randomization can circumvent this necessity is unknown, and the authors note that even in the simpler customer-only setting this question has remained unresolved for decades.