# Maximal value of directed graph with constraints

What is an algorithm to calculate the maximum "score" possible in a directed graph, with the constraint that edges with the same value can only be traversed once? For example, in the graph below, given that you can use each of a, b, and c once, the best path is to use a and b to get to 7, and c to get to 2, for a score of 9.

Edit: This is another way to formulate the problem:
Given: A set $S$ of $(key:value)$ pairs, where the key itself is a non-empty set of variables, eg - $S = \{(\{a\}:2), (\{c\}:2), (\{a, b\}:7), (\{b\}:3), (\{c\}:1)\}$. Keys may not be unique, so the word 'key' may be misleading.
Required: A subset $S' \subseteq S$ such that the total sum of values of each member of $S'$ is maximized.
Constraint: Each variable present in any of the keys of $S$ (eg. $a, b$ etc.) belongs to the list (key) of exactly one member of $S'$
For example, $S' = \{(\{c\}:2), (\{a, b\}:7)\}$ has a total value of 9, and each of $a, b, c$ occur in exactly one list.

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can you explain your problem in more detail ? how is the directed graph constructed ? and what do you want to obtain ? Tell us more about what is this problem applicable to. – AJed Jan 26 '13 at 2:26
Essentially, I have several workers that return proposals, that may be mutually exclusive. They each pertain to a single value. I.e.: "With a or c, I will provide 2." "With a and b I will provide 7." "With b, I will provide 3." "With c, I will provide 1." Except that, of course, in my real domain it will be more complex. – syrion Jan 26 '13 at 2:36
This is similar to the problem solved e.g. by Dijkstra's algorithm (just that looks for minimal cost, not maximal benefit). – vonbrand Jan 26 '13 at 3:47
From your example, all you have to do is to compares the values of the leaves $O(|V|)$. Find the leaf $l$ with highest score, and find the path from $s$ to $l$ .. $O(h)$ , where $h$ is height of the tree (btw: is it a tree ? or it can be any general DAG ?). Am I getting right ? – AJed Jan 26 '13 at 3:53
I am looking for a total score--probably unclear, but you start at S repeatedly until your keys are exhausted. – syrion Jan 26 '13 at 3:59