The Importance of Knowing the Arrival Order in Combinatorial Bayesian Settings

TL;DR

This paper investigates the order-competitive ratio in Bayesian combinatorial online algorithms, establishing lower bounds for various settings and demonstrating the limitations of improving approximation ratios without knowledge of arrival order.

Contribution

It provides novel constructions proving that certain lower bounds on approximation ratios hold even when considering the order-competitive ratio in combinatorial settings.

Findings

Lower bounds match previous benchmarks in single-choice and multi-unit settings.

Constant approximation is impossible for downward-closed constraints.

Achieving better than (1/\,n) approximation for general constraints is not feasible.

Abstract

We study the measure of order-competitive ratio introduced by Ezra et al. [2023] for online algorithms in Bayesian combinatorial settings. In our setting, a decision-maker observes a sequence of elements that are associated with stochastic rewards that are drawn from known priors, but revealed one by one in an online fashion. The decision-maker needs to decide upon the arrival of each element whether to select it or discard it (according to some feasibility constraint), and receives the associated rewards of the selected elements. The order-competitive ratio is defined as the worst-case ratio (over all distribution sequences) between the performance of the best order-unaware and order-aware algorithms, and quantifies the loss incurred due to the lack of knowledge of the arrival order. Ezra et al. [2023] showed how to design algorithms that achieve better approximations with respect to the new benchmark (order-competitive ratio) in the single-choice setting, which raises the natural question of whether the same can be achieved in combinatorial settings. In particular, whether it is possible to achieve a constant approximation with respect to the best online algorithm for downward-closed feasibility constraints, whether $\omega(1/n)$-approximation is achievable for general (non-downward-closed) feasibility constraints, or whether a convergence rate to $1$ of $o(1/\sqrt{k})$ is achievable for the multi-unit setting. We show, by devising novel constructions that may be of independent interest, that for all three scenarios, the asymptotic lower bounds with respect to the old benchmark, also hold with respect to the new benchmark.