Given edge labelled directed acyclic graph with edge labels $w_i \in \mathbb{N}$ the cost of a path is the sum of the labels.
The problem is:
Find a path from $s$ to $t$ with cost $a$.
I suppose this is NP-hard.
Some questions:
- Would putting a bound $\max w_i < C$ make the problem easier?
- What are the best algorithms for this?
- Is the problem NP-complete?
Update What about this modification of Number: 63 Shortest Weight-Constrained Path
To each edge associate second label weight $r_i \in \mathbb{N}$. The weight of a path is the sum of $r_i$.
Assume (possibly small) bound $C$ of the costs $w_i$ and solve:
$cost = a$ and $weight < K$.
Would it be still polynomial in $n,m,C$ or NP-hard instances exist (one can assume positive labels if this helps)?