  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
common external event. We then consider the Knapsack Secretary problem in this
model: there is a knapsack of size $k$ (e.g., available quantity of a good),
and in each of the $n$ time steps an item comes with its value and size in
$[0,1]$ and the algorithm needs to make an irrevocable decision whether to
accept or reject the item.
  We design an algorithm that gives an approximation of $1 -
\tilde{O}(\Gamma/k)$ when the adversarial time steps can be covered by $\Gamma
\ge \sqrt{k}$ intervals of size $\tilde{O}(\frac{n}{k})$. In particular,
setting $\Gamma = \sqrt{k}$ gives a $(1 - O(\frac{\ln^2
k}{\sqrt{k}}))$-approximation that is resistant to up to a $\frac{\ln^2
k}{\sqrt{k}}$-fraction of the items being adversarial, which is almost optimal
even in the absence of adversarial items. Also, setting $\Gamma =
\tilde{\Omega}(k)$ gives a constant approximation that is resistant to up to a
constant fraction of items being adversarial.
