123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223 |
- <HTML>
- <!--
- Copyright (c) Piotr Wygocki 2013
-
- Distributed under the Boost Software License, Version 1.0.
- (See accompanying file LICENSE_1_0.txt or copy at
- http://www.boost.org/LICENSE_1_0.txt)
- -->
- <Head>
- <Title>Boost Graph Library: Cycle Canceling for Min Cost Max Flow</Title>
- <BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
- ALINK="#ff0000">
- <IMG SRC="../../../boost.png"
- ALT="C++ Boost" width="277" height="86">
- <BR Clear>
- <H1><A NAME="sec:cycle_canceling">
- <TT>cycle_canceling</TT>
- </H1>
- <PRE>
- <i>// named parameter version</i>
- template <class <a href="./Graph.html">Graph</a>, class P, class T, class R>
- void cycle_canceling(
- Graph &g,
- const bgl_named_params<P, T, R> & params = <i>all defaults</i>)
- <i>// non-named parameter version</i>
- template <class <a href="./Graph.html">Graph</a>, class Pred, class Distance, class Reversed, class ResidualCapacity, class Weight>
- void cycle_canceling(const Graph & g, Weight weight, Reversed rev, ResidualCapacity residual_capacity, Pred pred, Distance distance)
- </PRE>
- <P>
- The <tt>cycle_canceling()</tt> function calculates the minimum cost flow of a network with given flow. See Section <a
- href="./graph_theory_review.html#sec:network-flow-algorithms">Network
- Flow Algorithms</a> for a description of maximum flow.
- For given flow values <i> f(u,v)</i> function minimizes flow cost in such a way, that for each <i>v in V</i> the
- <i> sum<sub> u in V</sub> f(v,u) </i> is preserved. Particularly if the input flow was the maximum flow, the function produces min cost max flow.
-
- The function calculates the flow values <i>f(u,v)</i> for all <i>(u,v)</i> in
- <i>E</i>, which are returned in the form of the residual capacity
- <i>r(u,v) = c(u,v) - f(u,v)</i>.
- <p>
- There are several special requirements on the input graph and property
- map parameters for this algorithm. First, the directed graph
- <i>G=(V,E)</i> that represents the network must be augmented to
- include the reverse edge for every edge in <i>E</i>. That is, the
- input graph should be <i>G<sub>in</sub> = (V,{E U
- E<sup>T</sup>})</i>. The <tt>ReverseEdgeMap</tt> argument <tt>rev</tt>
- must map each edge in the original graph to its reverse edge, that is
- <i>(u,v) -> (v,u)</i> for all <i>(u,v)</i> in <i>E</i>.
- The <tt>WeightMap</tt> has to map each edge from <i>E<sup>T</sup></i> to <i>-weight</i> of its reversed edge.
- Note that edges from <i>E</i> can have negative weights.
- <p>
- If weights in the graph are nonnegative, the
- <a href="./successive_shortest_path_nonnegative_weights.html"><tt>successive_shortest_path_nonnegative_weights()</tt></a>
- might be better choice for min cost max flow.
- <p>
- The algorithm is described in <a
- href="./bibliography.html#ahuja93:_network_flows">Network Flows</a>.
- <p>
- In each round algorithm augments the negative cycle (in terms of weight) in the residual graph.
- If there is no negative cycle in the network, the cost is optimized.
- <p>
- Note that, although we mention capacity in the problem description, the actual algorithm doesn't have to now it.
- <p>
- In order to find the cost of the result flow use:
- <a href="./find_flow_cost.html"><tt>find_flow_cost()</tt></a>.
- <H3>Where Defined</H3>
- <P>
- <a href="../../../boost/graph/successive_shortest_path_nonnegative_weights.hpp"><TT>boost/graph/successive_shortest_path_nonnegative_weights.hpp</TT></a>
- <P>
- <h3>Parameters</h3>
- IN: <tt>Graph& g</tt>
- <blockquote>
- A directed graph. The
- graph's type must be a model of <a
- href="./VertexListGraph.html">VertexListGraph</a> and <a href="./IncidenceGraph.html">IncidenceGraph</a> For each edge
- <i>(u,v)</i> in the graph, the reverse edge <i>(v,u)</i> must also
- be in the graph.
- </blockquote>
-
- <h3>Named Parameters</h3>
-
- IN/OUT: <tt>residual_capacity_map(ResidualCapacityEdgeMap res)</tt>
- <blockquote>
- This maps edges to their residual capacity. The type must be a model
- of a mutable <a
- href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property
- Map</a>. The key type of the map must be the graph's edge descriptor
- type.<br>
- <b>Default:</b> <tt>get(edge_residual_capacity, g)</tt>
- </blockquote>
- IN: <tt>reverse_edge_map(ReverseEdgeMap rev)</tt>
- <blockquote>
- An edge property map that maps every edge <i>(u,v)</i> in the graph
- to the reverse edge <i>(v,u)</i>. The map must be a model of
- constant <a href="../../property_map/doc/LvaluePropertyMap.html">Lvalue
- Property Map</a>. The key type of the map must be the graph's edge
- descriptor type.<br>
- <b>Default:</b> <tt>get(edge_reverse, g)</tt>
- </blockquote>
- IN: <tt>weight_map(WeightMap w)</tt>
- <blockquote>
- The weight (also know as ``length'' or ``cost'') of each edge in the
- graph. The <tt>WeightMap</tt> type must be a model of <a
- href="../../property_map/doc/ReadablePropertyMap.html">Readable Property
- Map</a>. The key type for this property map must be the edge
- descriptor of the graph. The value type for the weight map must be
- <i>Addable</i> with the distance map's value type. <br>
- <b>Default:</b> <tt>get(edge_weight, g)</tt><br>
- </blockquote>
- UTIL: <tt>predecessor_map(PredEdgeMap pred)</tt>
- <blockquote>
- Use by the algorithm to store augmenting paths. The map must be a
- model of mutable <a
- href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property Map</a>.
- The key type must be the graph's vertex descriptor type and the
- value type must be the graph's edge descriptor type.<br>
- <b>Default:</b> an <a
- href="../../property_map/doc/iterator_property_map.html">
- <tt>iterator_property_map</tt></a> created from a <tt>std::vector</tt>
- of edge descriptors of size <tt>num_vertices(g)</tt> and
- using the <tt>i_map</tt> for the index map.
- </blockquote>
- UTIL: <tt>distance_map(DistanceMap d_map)</tt>
- <blockquote>
- The shortest path weight from the source vertex <tt>s</tt> to each
- vertex in the graph <tt>g</tt> is recorded in this property map. The
- shortest path weight is the sum of the edge weights along the
- shortest path. The type <tt>DistanceMap</tt> must be a model of <a
- href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
- Property Map</a>. The vertex descriptor type of the graph needs to
- be usable as the key type of the distance map.
- <b>Default:</b> <a
- href="../../property_map/doc/iterator_property_map.html">
- <tt>iterator_property_map</tt></a> created from a
- <tt>std::vector</tt> of the <tt>WeightMap</tt>'s value type of size
- <tt>num_vertices(g)</tt> and using the <tt>i_map</tt> for the index
- map.<br>
- </blockquote>
- IN: <tt>vertex_index_map(VertexIndexMap i_map)</tt>
- <blockquote>
- Maps each vertex of the graph to a unique integer in the range
- <tt>[0, num_vertices(g))</tt>. This property map is only needed
- if the default for the distance or predecessor map is used.
- The vertex index map must be a model of <a
- href="../../property_map/doc/ReadablePropertyMap.html">Readable Property
- Map</a>. The key type of the map must be the graph's vertex
- descriptor type.<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>adjacency_list</tt> with <tt>VertexList=listS</tt> does
- not have an internal <tt>vertex_index</tt> property.
- </blockquote>
- <h3>Complexity</h3>
- In the integer capacity and weight case, if <i>C</i> is the initial cost of the flow, then the complexity is <i> O(C * |V| * |E|)</i>,
- where <i>O(|E|* |V|)</i> is the complexity of the bellman ford shortest paths algorithm and <i>C</i> is upper bound on number of iteration.
- In many real world cases number of iterations is much smaller than <i>C</i>.
- <h3>Example</h3>
- The program in <a
- href="../example/cycle_canceling_example.cpp"><tt>example/cycle_canceling_example.cpp</tt></a>.
- <h3>See Also</h3>
- <a href="./successive_shortest_path_nonnegative_weights.html"><tt>successive_shortest_path_nonnegative_weights()</tt></a><br>
- <a href="./find_flow_cost.html"><tt>find_flow_cost()</tt></a>.
- <br>
- <HR>
- <TABLE>
- <TR valign=top>
- <TD nowrap>Copyright © 2013</TD><TD>
- Piotr Wygocki, University of Warsaw (<A HREF="mailto:wygos@mimuw.edu.pl">wygos at mimuw.edu.pl</A>)
- </TD></TR></TABLE>
- </BODY>
- </HTML>
- <!-- LocalWords: HTML Siek Edmonds BGCOLOR ffffff ee VLINK ALINK ff IMG SRC
- -->
- <!-- LocalWords: gif ALT BR sec edmonds karp TT DIV CELLPADDING TR TD PRE lt
- -->
- <!-- LocalWords: typename VertexListGraph CapacityEdgeMap ReverseEdgeMap gt
- -->
- <!-- LocalWords: ResidualCapacityEdgeMap VertexIndexMap src rev ColorMap pred
- -->
- <!-- LocalWords: PredEdgeMap tt href html hpp ul li nbsp br LvaluePropertyMap
- -->
- <!-- LocalWords: num ColorValue DIMACS cpp pre config iostream dimacs int std
- -->
- <!-- LocalWords: namespace vecS directedS cout endl iter ei HR valign nowrap
- -->
- <!-- LocalWords: jeremy siek htm Univ mailto jsiek lsc edu
- p -->
|