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Consider a Directed Acyclic Graph in which every node has a value and a cost and edges do not have any weight. I need to find a path containing nodes such that sum of values of these nodes is maximized, but sum of cost over all these nodes must be at most C.
I am thinking of some modification of Dijkstra's algorithm. Any ideas? Is there any standard algorithm for this?
Consider a SUBSET SUM instance with weights $w_1,\ldots,w_n$ and target $C$. We create an instance of your problem, with the same value of $C$. There are $n+1$ nodes $v_1,\ldots,v_n$, where node $v_i$ has cost and weight $w_i$. There is an edge from $v_i$ to $v_j$ iff $i < j$.
Any directed path in this DAG corresponds to a subset of the weights $w_1,\ldots,w_n$. You are thus looking for a subset of the weights with maximal sum under the constraint that the sum is at most $C$. Thus the solution to your problem is $C$ iff the SUBSET SUM instance is a Yes instance.
