title: Visiting graph edges with independent paths
body: <p>Given a directed graph with multiple start nodes and multiple end nodes, I need to form paths that visit every reachable edge, but I cannot visit any edge (or vertex) more than once during a single pass.  [<em>This is to electrically test every connection in a network by sending signals from start to end nodes, but I cannot allow paths to short together.</em>]</p>
  
  <p>Because I cannot re-visit edges during a single pass:</p>
  
  <ul>
  <li>I can safely ignore the cycles in the graph.</li>
  <li>I know each path I form will block other paths.</li>
  <li>Consequently, I cannot visit every reachable edge in one pass, so multiple passes are necessary.</li>
  </ul>
  
  <p>From context, I know that the minimum number of passes will be the maximum number of edges entering any vertex.  Once I finish a given pass, I am free to re-visit edges that were visited in previous passes, but never-visited edges are the ones that I most want to visit.</p>
  
  <p>I would like to visit "<em>many</em>" edges per pass, so that I can reduce total the number of passes, but I do not strictly need to minimize the number of passes.</p>
  
  <p>Any suggestions on algorithms to accomplish this?  It sounds a little like the <a href="http://en.wikipedia.org/wiki/Route_inspection_problem" rel="nofollow">route inspection problem</a>, except that my graph is directed.</p>