Suppose I have 3 slices of bread, say $b_1$, $b_2$, $b_3$, and I want each to be double toasted. My toaster has a capacity of 2 slices at a time. The 'perfect' way of doing this would be to toast in the following manner - $(b_1,b_2),(b_2,b_3),(b_1,b_3)$. This 'perfect' strategy ensures that during each use of the toaster, I utilize both the toasting slots.
There does not exist a perfect strategy for toasting 3 slices of bread if I want my breads single toasted using the previous toaster.
Essentially, I want to know whether there exists a perfect strategy given that I have $T$ slices of bread, $K$ toasting slots in the toaster, and the vector $\vec V \in \mathbb{N}^T$ . Here, $\vec V$ describes the desired number of toasts for each bread.
If it is not clear, by perfect strategy I mean a strategy by which I will be utilizing all the slots of the toaster every time. Therefore, it is a 'yes/no' question. Perhaps, if possible, it would be great if we can also figure out the strategy as well.
I was thinking of this problem while toasting bread during breakfast and I have no idea as such on how to proceed!
If we have $T $ slices of bread and want the slices to be toastes $\vec V$ times, then if we had a single slot toaster, we'd need $\sum \vec V$ turns.
Since we have $K$ slots, we'll instead need at least $\frac{\sum \vec V}K$ turns.
Let's say that a strategy is perfect if it allows us to toast all toasts to the wished toastiness in $\lceil\frac{\sum \vec V}K\rceil$ turns.
Then we can find a perfect strategy for the toasting process, if one exists, by means of network flow (and therefore just as well using linear programming):
We create the graph as follows:
We create the source vertex $S_I$ and the sink vertex $S_O$.
We create a vertex $v_i$ for every toast.
We add a vertex $t_i$ for every turn.
For all $i$, we add an edge from the source $S_I$ to $v_i$ with capacity of $V_i$.
For all $i$ we add an edge from $t_i$ to the sink $S_O$ with capacity of $K$.
For all $i,j$ we add an edge with infinite capacity from $v_i$ to $t_j$
Now we search for a maximal flow in this graph. There's two outcomes:
Either we obtain a maximal flow that maximally utilizes all edges from $S_I$ to $v_i$. Then there exists a perfect strategy, and we con reconstruct it from the flow.
Or the maximal flow doesn't utilize all these edges maximally, in which case there is no perfect strategy.