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<Head>
<Title>Boost Graph Library: Metric Traveling Salesperson Approximation</Title>
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<H1><A NAME="sec:metric_tsp_approx"></A>
<TT>metric_tsp_approx</TT>
</H1>

<P>
<PRE>
template &lt;typename VertexListGraph,
          typename WeightMap,
          typename VertexIndexMap,
          typename TSPVertexVisitor&gt;
void metric_tsp_approx_from_vertex(
        const VertexListGraph&amp; g,
        typename graph_traits&lt;VertexListGraph&gt;::vertex_descriptor start,
        WeightMap weightmap,
        VertexIndexMap indexmap,
        TSPVertexVisitor vis)

<i>// Overloads</i>
template&lt;typename VertexListGraph, typename OutputIterator&gt;
void metric_tsp_approx_tour(VertexListGraph&amp; g, OutputIterator o)

template&lt;typename VertexListGraph, typename WeightMap, typename OutputIterator&gt;
void metric_tsp_approx_tour(VertexListGraph&amp; g, WeightMap w,
    OutputIterator o)

template&lt;typename VertexListGraph, typename OutputIterator&gt;
void metric_tsp_approx_tour_from_vertex(VertexListGraph&amp; g,
    typename graph_traits&lt;VertexListGraph&gt;::vertex_descriptor start,
    OutputIterator o)

template&lt;typename VertexListGraph, typename WeightMap,
    typename OutputIterator&gt;
void metric_tsp_approx_tour_from_vertex(VertexListGraph&amp; g,
    typename graph_traits&lt;VertexListGraph&gt;::vertex_descriptor start,
    WeightMap w, OutputIterator o)

<i>// visitor versions</i>
template&lt;typename VertexListGraph, typename TSPVertexVisitor&gt;
void metric_tsp_approx(VertexListGraph&amp; g, TSPVertexVisitor vis)

template&lt;typename VertexListGraph, typename Weightmap,
    typename VertexIndexMap, typename TSPVertexVisitor&gt;
void metric_tsp_approx(VertexListGraph&amp; g, Weightmap w,
    TSPVertexVisitor vis)

template&lt;typename VertexListGraph, typename WeightMap,
    typename VertexIndexMap, typename TSPVertexVisitor&gt;
void metric_tsp_approx(VertexListGraph&amp; g, WeightMap w, VertexIndexMap id,
    TSPVertexVisitor vis)

template&lt;typename VertexListGraph, typename WeightMap,
    typename TSPVertexVisitor&gt;
void metric_tsp_approx_from_vertex(VertexListGraph&amp; g,
    typename graph_traits&lt;VertexListGraph&gt;::vertex_descriptor start,
    WeightMap w, TSPVertexVisitor vis)

</PRE>

<P>
This is a traveling salesperson heuristic for generating a tour of vertices
for a fully connected undirected graph with weighted edges that obey the triangle inequality.
The current implementation is based off of the algorithm presented in <i>Introduction to Algorithms</i>
on page 1029.  This implementation generates solutions that are twice as long as the optimal tour in the worst case.
The pseudo-code is listed below.
</p>

<table>
<tr>
<td valign="top">
<pre>
TSP-APPROX(<i>G</i>, <i>w</i>)
  select a vertex <i>r</i> <i>in</i> <i>V[G]</i>
  compute a minimum spanning tree <i>T</i> for <i>G</i> from root <i>r</i>
    using PRIM-MST(<i>G</i>, <i>r</i>, <i>w</i>)
  let <i>L</i> be the list of vertices visited in a preorder traversal of <i>T</i>
  <b>return</b> (the Hamiltonian cycle <i>H</i> that visites the vertices in the order <i>L</i>)
</pre>
</td>
</tr>
</table>


<H3>Where Defined</H3>

<P>
<a href="../../../boost/graph/metric_tsp_approx.hpp"><TT>boost/graph/metric_tsp_approx.hpp</TT></a>

<P>

<h3>Parameters</h3>

IN: <tt>const Graph&amp; g</tt>
<blockquote>
  An undirected graph. The type <tt>Graph</tt> must be a
  model of <a href="./VertexListGraph.html">Vertex List Graph</a>
  and <a href="./IncidenceGraph.html">Incidence Graph</a> <a href="#2">[2]</a>.<br>
</blockquote>

IN: <tt>vertex_descriptor start</tt>
<blockquote>
  The vertex that will be the start (and end) of the tour.<br>
  <b>Default:</b> <tt>*vertices(g).first</tt>
</blockquote>

IN: <tt>WeightMap weightmap</tt>
<blockquote>
  The weight of each edge in the graph.
  The type <tt>WeightMap</tt> must be a model of
  <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The edge descriptor type of
  the graph needs to be usable as the key type for the weight
  map.<br>
  <b>Default:</b>  <tt>get(edge_weight, g)</tt><br>
</blockquote>

IN: <tt>VertexIndexMap indexmap</tt>
<blockquote>
  This maps each vertex to an integer in the range <tt>[0,
    num_vertices(g))</tt>. This is necessary for efficient updates of the
  heap data structure when an edge is relaxed.  The type
  <tt>VertexIndexMap</tt> must be a model of
  <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The value type of the map must be an
  integer type. The vertex descriptor type of the graph needs to be
  usable as the key type of the map.<br>
  <b>Default:</b> <tt>get(vertex_index, g)</tt>
    Note: if you use this default, make sure your graph has
    an internal <tt>vertex_index</tt> property. For example,
    <tt>adjacenty_list</tt> with <tt>VertexList=listS</tt> does
    not have an internal <tt>vertex_index</tt> property.
   <br>
</blockquote>

OUT: <tt>OutputIterator o</tt>
<blockquote>
  The OutputIterator holds the vertices of the tour.
  The type <tt>OutputIterator</tt> must be a model of the
  OutputIterator concept.
</blockquote>

OUT: <tt>TSPTourVisitor vis</tt>
<blockquote>
  Use this to specify actions that you would like to happen
  when the algorithm visits a vertex of the tour.
  The type <tt>TSPTourVisitor</tt> must be a model of the <a href="./TSPTourVisitor.html">TSP Tour Visitor</a>
  concept.
  The visitor object is passed by value <a
  href="#1">[1]</a>.<br>
</blockquote>

<H3>Complexity</H3>

<P>
The time complexity is <i>O(E log V)</i>.

<P>

<H3>Example</H3>

<P>
The file <a
href="../test/metric_tsp_approx.cpp"><TT>test/metric_tsp_approx.cpp</TT></a>
contains an example of using this TSP approximation algorithm.


<h3>Notes</h3>

<p><a name="1">[1]</a>
  Since the visitor parameter is passed by value, if your visitor
  contains state then any changes to the state during the algorithm
  will be made to a copy of the visitor object, not the visitor object
  passed in. Therefore you may want the visitor to hold this state by
  pointer or reference.
</p>
<p><a name="2">[2]</a>
  Passing an <tt>adjacency_list</tt> with a vertex <em>not</em> set selected by
  <tt>vecS</tt> will result in <i>O(n<sup>2</sup>)</i> performance.
</p>
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<TD nowrap>Copyright &copy; 2008</TD><TD>
Matyas Egyhazy
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