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<div class="section" id="network-simplex">
<h1>network_simplex<a class="headerlink" href="#network-simplex" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="networkx.algorithms.flow.network_simplex">
<tt class="descname">network_simplex</tt><big>(</big><em>G</em>, <em>demand='demand'</em>, <em>capacity='capacity'</em>, <em>weight='weight'</em><big>)</big><a class="headerlink" href="#networkx.algorithms.flow.network_simplex" title="Permalink to this definition">¶</a></dt>
<dd><p>Find a minimum cost flow satisfying all demands in digraph G.</p>
<p>This is a primal network simplex algorithm that uses the leaving
arc rule to prevent cycling.</p>
<p>G is a digraph with edge costs and capacities and in which nodes
have demand, i.e., they want to send or receive some amount of
flow. A negative demand means that the node wants to send flow, a
positive demand means that the node want to receive flow. A flow on
the digraph G satisfies all demand if the net flow into each node
is equal to the demand of that node.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>G</strong> : NetworkX graph</p>
<blockquote>
<div><p>DiGraph on which a minimum cost flow satisfying all demands is
to be found.</p>
</div></blockquote>
<p><strong>demand: string</strong> :</p>
<blockquote>
<div><p>Nodes of the graph G are expected to have an attribute demand
that indicates how much flow a node wants to send (negative
demand) or receive (positive demand). Note that the sum of the
demands should be 0 otherwise the problem in not feasible. If
this attribute is not present, a node is considered to have 0
demand. Default value: ‘demand’.</p>
</div></blockquote>
<p><strong>capacity: string</strong> :</p>
<blockquote>
<div><p>Edges of the graph G are expected to have an attribute capacity
that indicates how much flow the edge can support. If this
attribute is not present, the edge is considered to have
infinite capacity. Default value: ‘capacity’.</p>
</div></blockquote>
<p><strong>weight: string</strong> :</p>
<blockquote>
<div><p>Edges of the graph G are expected to have an attribute weight
that indicates the cost incurred by sending one unit of flow on
that edge. If not present, the weight is considered to be 0.
Default value: ‘weight’.</p>
</div></blockquote>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>flowCost: integer, float</strong> :</p>
<blockquote>
<div><p>Cost of a minimum cost flow satisfying all demands.</p>
</div></blockquote>
<p><strong>flowDict: dictionary</strong> :</p>
<blockquote>
<div><p>Dictionary of dictionaries keyed by nodes such that
flowDict[u][v] is the flow edge (u, v).</p>
</div></blockquote>
</td>
</tr>
<tr class="field-odd field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>NetworkXError</strong> :</p>
<blockquote>
<div><p>This exception is raised if the input graph is not directed,
not connected or is a multigraph.</p>
</div></blockquote>
<p><strong>NetworkXUnfeasible</strong> :</p>
<blockquote>
<div><dl class="docutils">
<dt>This exception is raised in the following situations:</dt>
<dd><ul class="first last simple">
<li>The sum of the demands is not zero. Then, there is no
flow satisfying all demands.</li>
<li>There is no flow satisfying all demand.</li>
</ul>
</dd>
</dl>
</div></blockquote>
<p><strong>NetworkXUnbounded</strong> :</p>
<blockquote class="last">
<div><p>This exception is raised if the digraph G has a cycle of
negative cost and infinite capacity. Then, the cost of a flow
satisfying all demands is unbounded below.</p>
</div></blockquote>
</td>
</tr>
</tbody>
</table>
<div class="admonition-see-also admonition seealso">
<p class="first admonition-title">See also</p>
<p class="last"><a class="reference internal" href="networkx.algorithms.flow.cost_of_flow.html#networkx.algorithms.flow.cost_of_flow" title="networkx.algorithms.flow.cost_of_flow"><tt class="xref py py-obj docutils literal"><span class="pre">cost_of_flow</span></tt></a>, <a class="reference internal" href="networkx.algorithms.flow.max_flow_min_cost.html#networkx.algorithms.flow.max_flow_min_cost" title="networkx.algorithms.flow.max_flow_min_cost"><tt class="xref py py-obj docutils literal"><span class="pre">max_flow_min_cost</span></tt></a>, <a class="reference internal" href="networkx.algorithms.flow.min_cost_flow.html#networkx.algorithms.flow.min_cost_flow" title="networkx.algorithms.flow.min_cost_flow"><tt class="xref py py-obj docutils literal"><span class="pre">min_cost_flow</span></tt></a>, <a class="reference internal" href="networkx.algorithms.flow.min_cost_flow_cost.html#networkx.algorithms.flow.min_cost_flow_cost" title="networkx.algorithms.flow.min_cost_flow_cost"><tt class="xref py py-obj docutils literal"><span class="pre">min_cost_flow_cost</span></tt></a></p>
</div>
<p class="rubric">Notes</p>
<p>This algorithm is not guaranteed to work if edge weights
are floating point numbers (overflows and roundoff errors can
cause problems).</p>
<p class="rubric">References</p>
<p>W. J. Cook, W. H. Cunningham, W. R. Pulleyblank and A. Schrijver.
Combinatorial Optimization. Wiley-Interscience, 1998.</p>
<p class="rubric">Examples</p>
<p>A simple example of a min cost flow problem.</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">networkx</span> <span class="kn">as</span> <span class="nn">nx</span>
<span class="gp">>>> </span><span class="n">G</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">DiGraph</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_node</span><span class="p">(</span><span class="s">'a'</span><span class="p">,</span> <span class="n">demand</span> <span class="o">=</span> <span class="o">-</span><span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_node</span><span class="p">(</span><span class="s">'d'</span><span class="p">,</span> <span class="n">demand</span> <span class="o">=</span> <span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="s">'a'</span><span class="p">,</span> <span class="s">'b'</span><span class="p">,</span> <span class="n">weight</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="n">capacity</span> <span class="o">=</span> <span class="mi">4</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="s">'a'</span><span class="p">,</span> <span class="s">'c'</span><span class="p">,</span> <span class="n">weight</span> <span class="o">=</span> <span class="mi">6</span><span class="p">,</span> <span class="n">capacity</span> <span class="o">=</span> <span class="mi">10</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="s">'b'</span><span class="p">,</span> <span class="s">'d'</span><span class="p">,</span> <span class="n">weight</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="n">capacity</span> <span class="o">=</span> <span class="mi">9</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="s">'c'</span><span class="p">,</span> <span class="s">'d'</span><span class="p">,</span> <span class="n">weight</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="n">capacity</span> <span class="o">=</span> <span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">flowCost</span><span class="p">,</span> <span class="n">flowDict</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">network_simplex</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">flowCost</span>
<span class="go">24</span>
<span class="gp">>>> </span><span class="n">flowDict</span>
<span class="go">{'a': {'c': 1, 'b': 4}, 'c': {'d': 1}, 'b': {'d': 4}, 'd': {}}</span>
</pre></div>
</div>
<p>The mincost flow algorithm can also be used to solve shortest path
problems. To find the shortest path between two nodes u and v,
give all edges an infinite capacity, give node u a demand of -1 and
node v a demand a 1. Then run the network simplex. The value of a
min cost flow will be the distance between u and v and edges
carrying positive flow will indicate the path.</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">G</span><span class="o">=</span><span class="n">nx</span><span class="o">.</span><span class="n">DiGraph</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_weighted_edges_from</span><span class="p">([(</span><span class="s">'s'</span><span class="p">,</span><span class="s">'u'</span><span class="p">,</span><span class="mi">10</span><span class="p">),</span> <span class="p">(</span><span class="s">'s'</span><span class="p">,</span><span class="s">'x'</span><span class="p">,</span><span class="mi">5</span><span class="p">),</span>
<span class="gp">... </span> <span class="p">(</span><span class="s">'u'</span><span class="p">,</span><span class="s">'v'</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="s">'u'</span><span class="p">,</span><span class="s">'x'</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span>
<span class="gp">... </span> <span class="p">(</span><span class="s">'v'</span><span class="p">,</span><span class="s">'y'</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="s">'x'</span><span class="p">,</span><span class="s">'u'</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span>
<span class="gp">... </span> <span class="p">(</span><span class="s">'x'</span><span class="p">,</span><span class="s">'v'</span><span class="p">,</span><span class="mi">5</span><span class="p">),</span> <span class="p">(</span><span class="s">'x'</span><span class="p">,</span><span class="s">'y'</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span>
<span class="gp">... </span> <span class="p">(</span><span class="s">'y'</span><span class="p">,</span><span class="s">'s'</span><span class="p">,</span><span class="mi">7</span><span class="p">),</span> <span class="p">(</span><span class="s">'y'</span><span class="p">,</span><span class="s">'v'</span><span class="p">,</span><span class="mi">6</span><span class="p">)])</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_node</span><span class="p">(</span><span class="s">'s'</span><span class="p">,</span> <span class="n">demand</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_node</span><span class="p">(</span><span class="s">'v'</span><span class="p">,</span> <span class="n">demand</span> <span class="o">=</span> <span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">flowCost</span><span class="p">,</span> <span class="n">flowDict</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">network_simplex</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">flowCost</span> <span class="o">==</span> <span class="n">nx</span><span class="o">.</span><span class="n">shortest_path_length</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="s">'s'</span><span class="p">,</span> <span class="s">'v'</span><span class="p">,</span> <span class="n">weight</span> <span class="o">=</span> <span class="s">'weight'</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="nb">sorted</span><span class="p">([(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">flowDict</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">flowDict</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="k">if</span> <span class="n">flowDict</span><span class="p">[</span><span class="n">u</span><span class="p">][</span><span class="n">v</span><span class="p">]</span> <span class="o">></span> <span class="mi">0</span><span class="p">])</span>
<span class="go">[('s', 'x'), ('u', 'v'), ('x', 'u')]</span>
<span class="gp">>>> </span><span class="n">nx</span><span class="o">.</span><span class="n">shortest_path</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="s">'s'</span><span class="p">,</span> <span class="s">'v'</span><span class="p">,</span> <span class="n">weight</span> <span class="o">=</span> <span class="s">'weight'</span><span class="p">)</span>
<span class="go">['s', 'x', 'u', 'v']</span>
</pre></div>
</div>
<p>It is possible to change the name of the attributes used for the
algorithm.</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">G</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">DiGraph</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_node</span><span class="p">(</span><span class="s">'p'</span><span class="p">,</span> <span class="n">spam</span> <span class="o">=</span> <span class="o">-</span><span class="mi">4</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_node</span><span class="p">(</span><span class="s">'q'</span><span class="p">,</span> <span class="n">spam</span> <span class="o">=</span> <span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_node</span><span class="p">(</span><span class="s">'a'</span><span class="p">,</span> <span class="n">spam</span> <span class="o">=</span> <span class="o">-</span><span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_node</span><span class="p">(</span><span class="s">'d'</span><span class="p">,</span> <span class="n">spam</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_node</span><span class="p">(</span><span class="s">'t'</span><span class="p">,</span> <span class="n">spam</span> <span class="o">=</span> <span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_node</span><span class="p">(</span><span class="s">'w'</span><span class="p">,</span> <span class="n">spam</span> <span class="o">=</span> <span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="s">'p'</span><span class="p">,</span> <span class="s">'q'</span><span class="p">,</span> <span class="n">cost</span> <span class="o">=</span> <span class="mi">7</span><span class="p">,</span> <span class="n">vacancies</span> <span class="o">=</span> <span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="s">'p'</span><span class="p">,</span> <span class="s">'a'</span><span class="p">,</span> <span class="n">cost</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="n">vacancies</span> <span class="o">=</span> <span class="mi">4</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="s">'q'</span><span class="p">,</span> <span class="s">'d'</span><span class="p">,</span> <span class="n">cost</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="n">vacancies</span> <span class="o">=</span> <span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="s">'t'</span><span class="p">,</span> <span class="s">'q'</span><span class="p">,</span> <span class="n">cost</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="n">vacancies</span> <span class="o">=</span> <span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="s">'a'</span><span class="p">,</span> <span class="s">'t'</span><span class="p">,</span> <span class="n">cost</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="n">vacancies</span> <span class="o">=</span> <span class="mi">4</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="s">'d'</span><span class="p">,</span> <span class="s">'w'</span><span class="p">,</span> <span class="n">cost</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="n">vacancies</span> <span class="o">=</span> <span class="mi">4</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">G</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="s">'t'</span><span class="p">,</span> <span class="s">'w'</span><span class="p">,</span> <span class="n">cost</span> <span class="o">=</span> <span class="mi">4</span><span class="p">,</span> <span class="n">vacancies</span> <span class="o">=</span> <span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">flowCost</span><span class="p">,</span> <span class="n">flowDict</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">network_simplex</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">demand</span> <span class="o">=</span> <span class="s">'spam'</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">capacity</span> <span class="o">=</span> <span class="s">'vacancies'</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">weight</span> <span class="o">=</span> <span class="s">'cost'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">flowCost</span>
<span class="go">37</span>
<span class="gp">>>> </span><span class="n">flowDict</span>
<span class="go">{'a': {'t': 4}, 'd': {'w': 2}, 'q': {'d': 1}, 'p': {'q': 2, 'a': 2}, 't': {'q': 1, 'w': 1}, 'w': {}}</span>
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