Randomization and the large-inventory threshold

Determine whether randomized online algorithms for the Competitive Bundle Trading problem—where a retailer buys from suppliers and sells to customers in bundles under per-item-type inventory caps—can achieve a finite competitive ratio without the large-inventory requirement that deterministic algorithms need; in particular, ascertain whether randomization can avoid the inventory-size restriction in the customer-only variant as well.

Background

The paper proves a threshold phenomenon for deterministic algorithms in the Competitive Bundle Trading model: to achieve a logarithmic competitive ratio, the per-type inventory capacity must be at least a logarithmic factor (in bundle size and valuation range) scaled by 1/epsilon. Moreover, the authors show that if inventory is below a smaller logarithmic threshold, no deterministic algorithm attains a finite competitive ratio.

This raises the question of whether randomization can fundamentally change the landscape. The authors note that even in the simpler customer-only setting—studied for decades—the status of whether randomized algorithms can bypass such inventory-size requirements has remained unresolved. The explicit open question asks whether randomization can avoid the large-inventory restriction in their full trading model and in the customer-only variant.