Transition coverage for a DFA Let $G$ be a directed graph, with a single source node $s$.  I want to find a collection of paths that cover every edge of $G$ (i.e., every edge of $G$ appears in at least one of these paths), where each path must start at $s$.  The cost of a collection of paths is the sum of the lengths of the paths.

Is there an efficient algorithm to find the minimum-cost collection of paths?  This smells like it might be NP-hard to me (it sounds like a set cover problem), so I'm guessing not.  If it is NP-hard, are there any good approximation algorithms or heuristics for this problem?

This sounds a bit like some kind of network-flow problem.  Every collection of paths from $s$ to $t$ is an integral flow from $s$ to $t$, and every integral flow from $s$ to $t$ can be expressed as a union of paths from $s$ to $t$.  The difference is that (1) the objective function we are trying to minimize is different from the standard network flow problem, and (2) the starting point of each edge is fixed to $s$, but the ending point is not fixed.  So, I don't know if network flow like techniques would help. 



Motivation: Suppose I have a DFA, and I want to find a testsuite that achieves full edge coverage.  A test is a word over the alphabet of the DFA; a testsuite is a collection of tests.  A transition is covered by this testsuite if there exists at least one test in the testsuite that causes the DFA to follow that transition at some point.  Suppose the cost of a test is its length, and the cost of a testsuite is the sum of the costs of the tests.

Then we can ask whether there is an efficient algorithm that, given a DFA as input, outputs a minimum-cost testsuite that achieves full transition coverage.  That's exactly the graph problem outlined above.
This is related to a well known problem with a conjecture due to Karpovsky and Moskalev [1] (I still believe is open) that at most V-1 paths are needed.  It is tied to conjector of Erdos, Goodman, and Posa [2].

If you want the state of the art today, I suggest doing a thorough reverse-citation search on [1].



[1] Karpovsky, M. G., and E. A. Moskalev. "Covering of edges of graph by a minimal set of paths." Discr. Math 58.2 (1986): 214. (This is in the "research problems" section)

[2] Erdos, Paul, Adolph W. Goodman, and Louis Pósa. "The representation of a graph by set intersections." Canad. J. Math 18.106-112 (1966): 86.