  A variant of the online knapsack problem is considered in the settings of
trusted and untrusted predictions. In Unit Profit Knapsack, the items have unit
profit, and it is easy to find an optimal solution offline: Pack as many of the
smallest items as possible into the knapsack. For Online Unit Profit Knapsack,
the competitive ratio is unbounded. In contrast, previous work on online
algorithms with untrusted predictions generally studied problems where an
online algorithm with a constant competitive ratio is known. The prediction,
possibly obtained from a machine learning source, that our algorithm uses is
the average size of those smallest items that fit in the knapsack. For the
prediction error in this hard online problem, we use the ratio
$r=\frac{a}{\hat{a}}$ where $a$ is the actual value for this average size and
$\hat{a}$ is the prediction. The algorithm presented achieves a competitive
ratio of $\frac{1}{2r}$ for $r\geq 1$ and $\frac{r}{2}$ for $r\leq 1$. Using an
adversary technique, we show that this is optimal in some sense, giving a
trade-off in the competitive ratio attainable for different values of $r$. Note
that the result for accurate advice, $r=1$, is only $\frac{1}{2}$, but we show
that no algorithm knowing the value $a$ can achieve a competitive ratio better
than $\frac{e-1}{e}\approx 0.6321$ and present an algorithm with a matching
upper bound. We also show that this latter algorithm attains a competitive
ratio of $r\frac{e-1}{e}$ for $r \leq 1$ and $\frac{e-r}{e}$ for $1 \leq r <
e$, and no algorithm can be better for both $r<1$ and $1\leq r<e$.
