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  1. PO
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    <p>Add all the "lower bounds" of the flows on each edge: any feasible solution will need that anyway.</p> <p>For each node <code>n</code> in the topological order (i.e. following the edges) from the sink <code>t</code>, if the sum <code>S-</code> of what goes in the edge is greater that the sum <code>S+</code> of what goes out, then add the flow <code>S+ - S-</code> on all edges of the shortest path between <code>s</code> and <code>t</code> (construct the list of shortest path while doing this for efficiency).</p> <p>Then, you have a 'minimal' assigment (in the sense that it is needed in every feasible solution) such as <code>S+ - S-</code> is nonnegative at each node and the minimal capacity is satisfied on each edge.</p> <p>The problem then reduce to a multiflow problem on the same graph structure:</p> <ul> <li>the source is the same;</li> <li>there are no capacity limit;</li> <li>every node where <code>S+ - S-</code> is strictly positive becomes a sink with required flow <code>S+ - S-</code>.</li> <li>the initial sink (which had a required flow <code>F</code>) becomes a sink with flow <code>F - S-</code> if <code>F-S-</code> is nonnegative (otherwise the initial constraint will be satisfied by any solution).</li> </ul> <p>Edit: for your problem, the graph looks like this</p> <pre><code> x1 -- x2 / \ \ s \ t \ \ / x3 -- x4 </code></pre> <p>with minimal capacities 1 for each edge, and I suppose there is no minimal flow required at the sink <code>t</code>.</p> <p>First assign 1 to each edge.</p> <p>The difference <code>S+ - S-</code> for each node (except, of course, <code>s</code> and <code>t</code>) is:</p> <pre><code>x1: 1 x2: 0 x3: 0 x4: -1 </code></pre> <p>the only negative one is for <code>x4</code>; we add <code>1</code> to every edge on the shortest path from <code>x4</code> to <code>t</code>, in that case to the edge <code>(x4, t)</code>.</p> <p>Now <code>S+ - S-</code> is non-negative for every node, and positive only for <code>x1</code>; the problem reduce to a multiflow problem (in that case it is a simple flow problem) where the requested flow is <code>1</code> at <code>x1</code>, and the source is still <code>s</code>.</p>
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    1. POWhat algorithm should I use to find the minimum flow on a digraph where there are lower bounds but not upper bounds on flow?
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    1. PTAnswer
    1. USNicolas Grebille
    1. USNicolas Grebille
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    1. COSo f is the flow assignment in which the flow across each edge is just equal to the lower bound?
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      1. PO
      1. USjwezorek
    2. COf will not be a legal flow meaning for a given node the incoming flow will not equal the outgoing flow. Therefore merely augmenting f along the shortest path from s to t is not going to necessarily result in a legal flow. What about nodes of f where conservation of flow is not satisfied that are not on the shortest path, for example.
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      1. USjwezorek
    3. COYes you are right. I try to adapt the solution right now, I will edit the answer if it works.
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      1. USNicolas Grebille
 

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