Knapsack Secretary with Bursty Adversary
Thomas Kesselheim, Marco Molinaro

TL;DR
This paper introduces a robust online algorithm for the Knapsack Secretary problem that performs well even when adversarial items appear in bursts, bridging stochastic and adversarial input models.
Contribution
It proposes a new model with bursty adversarial behavior and designs an algorithm achieving near-optimal approximation ratios under this model.
Findings
Achieves a $(1 - O(rac{ ext{polylog}(k)}{ ootk}))$-approximation with burst size $ ilde{O}(rac{n}{k})$
Resistant to a $rac{ ext{polylog}(k)}{ ootk}$ fraction of adversarial items
Provides a constant-factor approximation when adversarial bursts are proportional to $k$
Abstract
The random-order or secretary model is one of the most popular beyond-worst case model for online algorithms. While it avoids the pessimism of the traditional adversarial model, in practice we cannot expect the input to be presented in perfectly random order. This has motivated research on ``best of both worlds'' (algorithms with good performance on both purely stochastic and purely adversarial inputs), or even better, on inputs that are a mix of both stochastic and adversarial parts. Unfortunately the latter seems much harder to achieve and very few results of this type are known. Towards advancing our understanding of designing such robust algorithms, we propose a random-order model with bursts of adversarial time steps. The assumption of burstiness of unexpected patterns is reasonable in many contexts, since changes (e.g. spike in a demand for a good) are often triggered by a…
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