Path of maximum value with bounded cost in DAG
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 $i &lt; j$.

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.