doc/groups.dox
author Peter Kovacs <kpeter@inf.elte.hu>
Wed, 29 Apr 2009 03:15:24 +0200
changeset 687 6c408d864fa1
parent 658 85cb3aa71cce
child 698 3adf5e2d1e62
child 843 189760a7cdd0
permissions -rw-r--r--
Support negative costs and bounds in NetworkSimplex (#270)

* The interface is reworked to support negative costs and bounds.
- ProblemType and problemType() are renamed to
SupplyType and supplyType(), see also #234.
- ProblemType type is introduced similarly to the LP interface.
- 'bool run()' is replaced by 'ProblemType run()' to handle
unbounded problem instances, as well.
- Add INF public member constant similarly to the LP interface.
* Remove capacityMap() and boundMaps(), see also #266.
* Update the problem definition in the MCF module.
* Remove the usage of Circulation (and adaptors) for checking feasibility.
Check feasibility by examining the artifical arcs instead (after solving
the problem).
* Additional check for unbounded negative cycles found during the
algorithm (it is possible now, since negative costs are allowed).
* Fix in the constructor (the value types needn't be integer any more),
see also #254.
* Improve and extend the doc.
* Rework the test file and add test cases for negative costs and bounds.
     1 /* -*- mode: C++; indent-tabs-mode: nil; -*-
     2  *
     3  * This file is a part of LEMON, a generic C++ optimization library.
     4  *
     5  * Copyright (C) 2003-2009
     6  * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
     7  * (Egervary Research Group on Combinatorial Optimization, EGRES).
     8  *
     9  * Permission to use, modify and distribute this software is granted
    10  * provided that this copyright notice appears in all copies. For
    11  * precise terms see the accompanying LICENSE file.
    12  *
    13  * This software is provided "AS IS" with no warranty of any kind,
    14  * express or implied, and with no claim as to its suitability for any
    15  * purpose.
    16  *
    17  */
    18 
    19 namespace lemon {
    20 
    21 /**
    22 @defgroup datas Data Structures
    23 This group contains the several data structures implemented in LEMON.
    24 */
    25 
    26 /**
    27 @defgroup graphs Graph Structures
    28 @ingroup datas
    29 \brief Graph structures implemented in LEMON.
    30 
    31 The implementation of combinatorial algorithms heavily relies on
    32 efficient graph implementations. LEMON offers data structures which are
    33 planned to be easily used in an experimental phase of implementation studies,
    34 and thereafter the program code can be made efficient by small modifications.
    35 
    36 The most efficient implementation of diverse applications require the
    37 usage of different physical graph implementations. These differences
    38 appear in the size of graph we require to handle, memory or time usage
    39 limitations or in the set of operations through which the graph can be
    40 accessed.  LEMON provides several physical graph structures to meet
    41 the diverging requirements of the possible users.  In order to save on
    42 running time or on memory usage, some structures may fail to provide
    43 some graph features like arc/edge or node deletion.
    44 
    45 Alteration of standard containers need a very limited number of
    46 operations, these together satisfy the everyday requirements.
    47 In the case of graph structures, different operations are needed which do
    48 not alter the physical graph, but gives another view. If some nodes or
    49 arcs have to be hidden or the reverse oriented graph have to be used, then
    50 this is the case. It also may happen that in a flow implementation
    51 the residual graph can be accessed by another algorithm, or a node-set
    52 is to be shrunk for another algorithm.
    53 LEMON also provides a variety of graphs for these requirements called
    54 \ref graph_adaptors "graph adaptors". Adaptors cannot be used alone but only
    55 in conjunction with other graph representations.
    56 
    57 You are free to use the graph structure that fit your requirements
    58 the best, most graph algorithms and auxiliary data structures can be used
    59 with any graph structure.
    60 
    61 <b>See also:</b> \ref graph_concepts "Graph Structure Concepts".
    62 */
    63 
    64 /**
    65 @defgroup graph_adaptors Adaptor Classes for Graphs
    66 @ingroup graphs
    67 \brief Adaptor classes for digraphs and graphs
    68 
    69 This group contains several useful adaptor classes for digraphs and graphs.
    70 
    71 The main parts of LEMON are the different graph structures, generic
    72 graph algorithms, graph concepts, which couple them, and graph
    73 adaptors. While the previous notions are more or less clear, the
    74 latter one needs further explanation. Graph adaptors are graph classes
    75 which serve for considering graph structures in different ways.
    76 
    77 A short example makes this much clearer.  Suppose that we have an
    78 instance \c g of a directed graph type, say ListDigraph and an algorithm
    79 \code
    80 template <typename Digraph>
    81 int algorithm(const Digraph&);
    82 \endcode
    83 is needed to run on the reverse oriented graph.  It may be expensive
    84 (in time or in memory usage) to copy \c g with the reversed
    85 arcs.  In this case, an adaptor class is used, which (according
    86 to LEMON \ref concepts::Digraph "digraph concepts") works as a digraph.
    87 The adaptor uses the original digraph structure and digraph operations when
    88 methods of the reversed oriented graph are called.  This means that the adaptor
    89 have minor memory usage, and do not perform sophisticated algorithmic
    90 actions.  The purpose of it is to give a tool for the cases when a
    91 graph have to be used in a specific alteration.  If this alteration is
    92 obtained by a usual construction like filtering the node or the arc set or
    93 considering a new orientation, then an adaptor is worthwhile to use.
    94 To come back to the reverse oriented graph, in this situation
    95 \code
    96 template<typename Digraph> class ReverseDigraph;
    97 \endcode
    98 template class can be used. The code looks as follows
    99 \code
   100 ListDigraph g;
   101 ReverseDigraph<ListDigraph> rg(g);
   102 int result = algorithm(rg);
   103 \endcode
   104 During running the algorithm, the original digraph \c g is untouched.
   105 This techniques give rise to an elegant code, and based on stable
   106 graph adaptors, complex algorithms can be implemented easily.
   107 
   108 In flow, circulation and matching problems, the residual
   109 graph is of particular importance. Combining an adaptor implementing
   110 this with shortest path algorithms or minimum mean cycle algorithms,
   111 a range of weighted and cardinality optimization algorithms can be
   112 obtained. For other examples, the interested user is referred to the
   113 detailed documentation of particular adaptors.
   114 
   115 The behavior of graph adaptors can be very different. Some of them keep
   116 capabilities of the original graph while in other cases this would be
   117 meaningless. This means that the concepts that they meet depend
   118 on the graph adaptor, and the wrapped graph.
   119 For example, if an arc of a reversed digraph is deleted, this is carried
   120 out by deleting the corresponding arc of the original digraph, thus the
   121 adaptor modifies the original digraph.
   122 However in case of a residual digraph, this operation has no sense.
   123 
   124 Let us stand one more example here to simplify your work.
   125 ReverseDigraph has constructor
   126 \code
   127 ReverseDigraph(Digraph& digraph);
   128 \endcode
   129 This means that in a situation, when a <tt>const %ListDigraph&</tt>
   130 reference to a graph is given, then it have to be instantiated with
   131 <tt>Digraph=const %ListDigraph</tt>.
   132 \code
   133 int algorithm1(const ListDigraph& g) {
   134   ReverseDigraph<const ListDigraph> rg(g);
   135   return algorithm2(rg);
   136 }
   137 \endcode
   138 */
   139 
   140 /**
   141 @defgroup semi_adaptors Semi-Adaptor Classes for Graphs
   142 @ingroup graphs
   143 \brief Graph types between real graphs and graph adaptors.
   144 
   145 This group contains some graph types between real graphs and graph adaptors.
   146 These classes wrap graphs to give new functionality as the adaptors do it.
   147 On the other hand they are not light-weight structures as the adaptors.
   148 */
   149 
   150 /**
   151 @defgroup maps Maps
   152 @ingroup datas
   153 \brief Map structures implemented in LEMON.
   154 
   155 This group contains the map structures implemented in LEMON.
   156 
   157 LEMON provides several special purpose maps and map adaptors that e.g. combine
   158 new maps from existing ones.
   159 
   160 <b>See also:</b> \ref map_concepts "Map Concepts".
   161 */
   162 
   163 /**
   164 @defgroup graph_maps Graph Maps
   165 @ingroup maps
   166 \brief Special graph-related maps.
   167 
   168 This group contains maps that are specifically designed to assign
   169 values to the nodes and arcs/edges of graphs.
   170 
   171 If you are looking for the standard graph maps (\c NodeMap, \c ArcMap,
   172 \c EdgeMap), see the \ref graph_concepts "Graph Structure Concepts".
   173 */
   174 
   175 /**
   176 \defgroup map_adaptors Map Adaptors
   177 \ingroup maps
   178 \brief Tools to create new maps from existing ones
   179 
   180 This group contains map adaptors that are used to create "implicit"
   181 maps from other maps.
   182 
   183 Most of them are \ref concepts::ReadMap "read-only maps".
   184 They can make arithmetic and logical operations between one or two maps
   185 (negation, shifting, addition, multiplication, logical 'and', 'or',
   186 'not' etc.) or e.g. convert a map to another one of different Value type.
   187 
   188 The typical usage of this classes is passing implicit maps to
   189 algorithms.  If a function type algorithm is called then the function
   190 type map adaptors can be used comfortable. For example let's see the
   191 usage of map adaptors with the \c graphToEps() function.
   192 \code
   193   Color nodeColor(int deg) {
   194     if (deg >= 2) {
   195       return Color(0.5, 0.0, 0.5);
   196     } else if (deg == 1) {
   197       return Color(1.0, 0.5, 1.0);
   198     } else {
   199       return Color(0.0, 0.0, 0.0);
   200     }
   201   }
   202 
   203   Digraph::NodeMap<int> degree_map(graph);
   204 
   205   graphToEps(graph, "graph.eps")
   206     .coords(coords).scaleToA4().undirected()
   207     .nodeColors(composeMap(functorToMap(nodeColor), degree_map))
   208     .run();
   209 \endcode
   210 The \c functorToMap() function makes an \c int to \c Color map from the
   211 \c nodeColor() function. The \c composeMap() compose the \c degree_map
   212 and the previously created map. The composed map is a proper function to
   213 get the color of each node.
   214 
   215 The usage with class type algorithms is little bit harder. In this
   216 case the function type map adaptors can not be used, because the
   217 function map adaptors give back temporary objects.
   218 \code
   219   Digraph graph;
   220 
   221   typedef Digraph::ArcMap<double> DoubleArcMap;
   222   DoubleArcMap length(graph);
   223   DoubleArcMap speed(graph);
   224 
   225   typedef DivMap<DoubleArcMap, DoubleArcMap> TimeMap;
   226   TimeMap time(length, speed);
   227 
   228   Dijkstra<Digraph, TimeMap> dijkstra(graph, time);
   229   dijkstra.run(source, target);
   230 \endcode
   231 We have a length map and a maximum speed map on the arcs of a digraph.
   232 The minimum time to pass the arc can be calculated as the division of
   233 the two maps which can be done implicitly with the \c DivMap template
   234 class. We use the implicit minimum time map as the length map of the
   235 \c Dijkstra algorithm.
   236 */
   237 
   238 /**
   239 @defgroup matrices Matrices
   240 @ingroup datas
   241 \brief Two dimensional data storages implemented in LEMON.
   242 
   243 This group contains two dimensional data storages implemented in LEMON.
   244 */
   245 
   246 /**
   247 @defgroup paths Path Structures
   248 @ingroup datas
   249 \brief %Path structures implemented in LEMON.
   250 
   251 This group contains the path structures implemented in LEMON.
   252 
   253 LEMON provides flexible data structures to work with paths.
   254 All of them have similar interfaces and they can be copied easily with
   255 assignment operators and copy constructors. This makes it easy and
   256 efficient to have e.g. the Dijkstra algorithm to store its result in
   257 any kind of path structure.
   258 
   259 \sa lemon::concepts::Path
   260 */
   261 
   262 /**
   263 @defgroup auxdat Auxiliary Data Structures
   264 @ingroup datas
   265 \brief Auxiliary data structures implemented in LEMON.
   266 
   267 This group contains some data structures implemented in LEMON in
   268 order to make it easier to implement combinatorial algorithms.
   269 */
   270 
   271 /**
   272 @defgroup algs Algorithms
   273 \brief This group contains the several algorithms
   274 implemented in LEMON.
   275 
   276 This group contains the several algorithms
   277 implemented in LEMON.
   278 */
   279 
   280 /**
   281 @defgroup search Graph Search
   282 @ingroup algs
   283 \brief Common graph search algorithms.
   284 
   285 This group contains the common graph search algorithms, namely
   286 \e breadth-first \e search (BFS) and \e depth-first \e search (DFS).
   287 */
   288 
   289 /**
   290 @defgroup shortest_path Shortest Path Algorithms
   291 @ingroup algs
   292 \brief Algorithms for finding shortest paths.
   293 
   294 This group contains the algorithms for finding shortest paths in digraphs.
   295 
   296  - \ref Dijkstra algorithm for finding shortest paths from a source node
   297    when all arc lengths are non-negative.
   298  - \ref BellmanFord "Bellman-Ford" algorithm for finding shortest paths
   299    from a source node when arc lenghts can be either positive or negative,
   300    but the digraph should not contain directed cycles with negative total
   301    length.
   302  - \ref FloydWarshall "Floyd-Warshall" and \ref Johnson "Johnson" algorithms
   303    for solving the \e all-pairs \e shortest \e paths \e problem when arc
   304    lenghts can be either positive or negative, but the digraph should
   305    not contain directed cycles with negative total length.
   306  - \ref Suurballe A successive shortest path algorithm for finding
   307    arc-disjoint paths between two nodes having minimum total length.
   308 */
   309 
   310 /**
   311 @defgroup max_flow Maximum Flow Algorithms
   312 @ingroup algs
   313 \brief Algorithms for finding maximum flows.
   314 
   315 This group contains the algorithms for finding maximum flows and
   316 feasible circulations.
   317 
   318 The \e maximum \e flow \e problem is to find a flow of maximum value between
   319 a single source and a single target. Formally, there is a \f$G=(V,A)\f$
   320 digraph, a \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function and
   321 \f$s, t \in V\f$ source and target nodes.
   322 A maximum flow is an \f$f: A\rightarrow\mathbf{R}^+_0\f$ solution of the
   323 following optimization problem.
   324 
   325 \f[ \max\sum_{sv\in A} f(sv) - \sum_{vs\in A} f(vs) \f]
   326 \f[ \sum_{uv\in A} f(uv) = \sum_{vu\in A} f(vu)
   327     \quad \forall u\in V\setminus\{s,t\} \f]
   328 \f[ 0 \leq f(uv) \leq cap(uv) \quad \forall uv\in A \f]
   329 
   330 LEMON contains several algorithms for solving maximum flow problems:
   331 - \ref EdmondsKarp Edmonds-Karp algorithm.
   332 - \ref Preflow Goldberg-Tarjan's preflow push-relabel algorithm.
   333 - \ref DinitzSleatorTarjan Dinitz's blocking flow algorithm with dynamic trees.
   334 - \ref GoldbergTarjan Preflow push-relabel algorithm with dynamic trees.
   335 
   336 In most cases the \ref Preflow "Preflow" algorithm provides the
   337 fastest method for computing a maximum flow. All implementations
   338 provides functions to also query the minimum cut, which is the dual
   339 problem of the maximum flow.
   340 */
   341 
   342 /**
   343 @defgroup min_cost_flow Minimum Cost Flow Algorithms
   344 @ingroup algs
   345 
   346 \brief Algorithms for finding minimum cost flows and circulations.
   347 
   348 This group contains the algorithms for finding minimum cost flows and
   349 circulations.
   350 
   351 The \e minimum \e cost \e flow \e problem is to find a feasible flow of
   352 minimum total cost from a set of supply nodes to a set of demand nodes
   353 in a network with capacity constraints (lower and upper bounds)
   354 and arc costs.
   355 Formally, let \f$G=(V,A)\f$ be a digraph, \f$lower: A\rightarrow\mathbf{Z}\f$,
   356 \f$upper: A\rightarrow\mathbf{Z}\cup\{+\infty\}\f$ denote the lower and
   357 upper bounds for the flow values on the arcs, for which
   358 \f$lower(uv) \leq upper(uv)\f$ must hold for all \f$uv\in A\f$,
   359 \f$cost: A\rightarrow\mathbf{Z}\f$ denotes the cost per unit flow
   360 on the arcs and \f$sup: V\rightarrow\mathbf{Z}\f$ denotes the
   361 signed supply values of the nodes.
   362 If \f$sup(u)>0\f$, then \f$u\f$ is a supply node with \f$sup(u)\f$
   363 supply, if \f$sup(u)<0\f$, then \f$u\f$ is a demand node with
   364 \f$-sup(u)\f$ demand.
   365 A minimum cost flow is an \f$f: A\rightarrow\mathbf{Z}\f$ solution
   366 of the following optimization problem.
   367 
   368 \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f]
   369 \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \geq
   370     sup(u) \quad \forall u\in V \f]
   371 \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f]
   372 
   373 The sum of the supply values, i.e. \f$\sum_{u\in V} sup(u)\f$ must be
   374 zero or negative in order to have a feasible solution (since the sum
   375 of the expressions on the left-hand side of the inequalities is zero).
   376 It means that the total demand must be greater or equal to the total
   377 supply and all the supplies have to be carried out from the supply nodes,
   378 but there could be demands that are not satisfied.
   379 If \f$\sum_{u\in V} sup(u)\f$ is zero, then all the supply/demand
   380 constraints have to be satisfied with equality, i.e. all demands
   381 have to be satisfied and all supplies have to be used.
   382 
   383 If you need the opposite inequalities in the supply/demand constraints
   384 (i.e. the total demand is less than the total supply and all the demands
   385 have to be satisfied while there could be supplies that are not used),
   386 then you could easily transform the problem to the above form by reversing
   387 the direction of the arcs and taking the negative of the supply values
   388 (e.g. using \ref ReverseDigraph and \ref NegMap adaptors).
   389 However \ref NetworkSimplex algorithm also supports this form directly
   390 for the sake of convenience.
   391 
   392 A feasible solution for this problem can be found using \ref Circulation.
   393 
   394 Note that the above formulation is actually more general than the usual
   395 definition of the minimum cost flow problem, in which strict equalities
   396 are required in the supply/demand contraints, i.e.
   397 
   398 \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) =
   399     sup(u) \quad \forall u\in V. \f]
   400 
   401 However if the sum of the supply values is zero, then these two problems
   402 are equivalent. So if you need the equality form, you have to ensure this
   403 additional contraint for the algorithms.
   404 
   405 The dual solution of the minimum cost flow problem is represented by node 
   406 potentials \f$\pi: V\rightarrow\mathbf{Z}\f$.
   407 An \f$f: A\rightarrow\mathbf{Z}\f$ feasible solution of the problem
   408 is optimal if and only if for some \f$\pi: V\rightarrow\mathbf{Z}\f$
   409 node potentials the following \e complementary \e slackness optimality
   410 conditions hold.
   411 
   412  - For all \f$uv\in A\f$ arcs:
   413    - if \f$cost^\pi(uv)>0\f$, then \f$f(uv)=lower(uv)\f$;
   414    - if \f$lower(uv)<f(uv)<upper(uv)\f$, then \f$cost^\pi(uv)=0\f$;
   415    - if \f$cost^\pi(uv)<0\f$, then \f$f(uv)=upper(uv)\f$.
   416  - For all \f$u\in V\f$ nodes:
   417    - if \f$\sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \neq sup(u)\f$,
   418      then \f$\pi(u)=0\f$.
   419  
   420 Here \f$cost^\pi(uv)\f$ denotes the \e reduced \e cost of the arc
   421 \f$uv\in A\f$ with respect to the potential function \f$\pi\f$, i.e.
   422 \f[ cost^\pi(uv) = cost(uv) + \pi(u) - \pi(v).\f]
   423 
   424 All algorithms provide dual solution (node potentials) as well,
   425 if an optimal flow is found.
   426 
   427 LEMON contains several algorithms for solving minimum cost flow problems.
   428  - \ref NetworkSimplex Primal Network Simplex algorithm with various
   429    pivot strategies.
   430  - \ref CostScaling Push-Relabel and Augment-Relabel algorithms based on
   431    cost scaling.
   432  - \ref CapacityScaling Successive Shortest %Path algorithm with optional
   433    capacity scaling.
   434  - \ref CancelAndTighten The Cancel and Tighten algorithm.
   435  - \ref CycleCanceling Cycle-Canceling algorithms.
   436 
   437 Most of these implementations support the general inequality form of the
   438 minimum cost flow problem, but CancelAndTighten and CycleCanceling
   439 only support the equality form due to the primal method they use.
   440 
   441 In general NetworkSimplex is the most efficient implementation,
   442 but in special cases other algorithms could be faster.
   443 For example, if the total supply and/or capacities are rather small,
   444 CapacityScaling is usually the fastest algorithm (without effective scaling).
   445 */
   446 
   447 /**
   448 @defgroup min_cut Minimum Cut Algorithms
   449 @ingroup algs
   450 
   451 \brief Algorithms for finding minimum cut in graphs.
   452 
   453 This group contains the algorithms for finding minimum cut in graphs.
   454 
   455 The \e minimum \e cut \e problem is to find a non-empty and non-complete
   456 \f$X\f$ subset of the nodes with minimum overall capacity on
   457 outgoing arcs. Formally, there is a \f$G=(V,A)\f$ digraph, a
   458 \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function. The minimum
   459 cut is the \f$X\f$ solution of the next optimization problem:
   460 
   461 \f[ \min_{X \subset V, X\not\in \{\emptyset, V\}}
   462     \sum_{uv\in A, u\in X, v\not\in X}cap(uv) \f]
   463 
   464 LEMON contains several algorithms related to minimum cut problems:
   465 
   466 - \ref HaoOrlin "Hao-Orlin algorithm" for calculating minimum cut
   467   in directed graphs.
   468 - \ref NagamochiIbaraki "Nagamochi-Ibaraki algorithm" for
   469   calculating minimum cut in undirected graphs.
   470 - \ref GomoryHu "Gomory-Hu tree computation" for calculating
   471   all-pairs minimum cut in undirected graphs.
   472 
   473 If you want to find minimum cut just between two distinict nodes,
   474 see the \ref max_flow "maximum flow problem".
   475 */
   476 
   477 /**
   478 @defgroup graph_properties Connectivity and Other Graph Properties
   479 @ingroup algs
   480 \brief Algorithms for discovering the graph properties
   481 
   482 This group contains the algorithms for discovering the graph properties
   483 like connectivity, bipartiteness, euler property, simplicity etc.
   484 
   485 \image html edge_biconnected_components.png
   486 \image latex edge_biconnected_components.eps "bi-edge-connected components" width=\textwidth
   487 */
   488 
   489 /**
   490 @defgroup planar Planarity Embedding and Drawing
   491 @ingroup algs
   492 \brief Algorithms for planarity checking, embedding and drawing
   493 
   494 This group contains the algorithms for planarity checking,
   495 embedding and drawing.
   496 
   497 \image html planar.png
   498 \image latex planar.eps "Plane graph" width=\textwidth
   499 */
   500 
   501 /**
   502 @defgroup matching Matching Algorithms
   503 @ingroup algs
   504 \brief Algorithms for finding matchings in graphs and bipartite graphs.
   505 
   506 This group contains the algorithms for calculating
   507 matchings in graphs and bipartite graphs. The general matching problem is
   508 finding a subset of the edges for which each node has at most one incident
   509 edge.
   510 
   511 There are several different algorithms for calculate matchings in
   512 graphs.  The matching problems in bipartite graphs are generally
   513 easier than in general graphs. The goal of the matching optimization
   514 can be finding maximum cardinality, maximum weight or minimum cost
   515 matching. The search can be constrained to find perfect or
   516 maximum cardinality matching.
   517 
   518 The matching algorithms implemented in LEMON:
   519 - \ref MaxBipartiteMatching Hopcroft-Karp augmenting path algorithm
   520   for calculating maximum cardinality matching in bipartite graphs.
   521 - \ref PrBipartiteMatching Push-relabel algorithm
   522   for calculating maximum cardinality matching in bipartite graphs.
   523 - \ref MaxWeightedBipartiteMatching
   524   Successive shortest path algorithm for calculating maximum weighted
   525   matching and maximum weighted bipartite matching in bipartite graphs.
   526 - \ref MinCostMaxBipartiteMatching
   527   Successive shortest path algorithm for calculating minimum cost maximum
   528   matching in bipartite graphs.
   529 - \ref MaxMatching Edmond's blossom shrinking algorithm for calculating
   530   maximum cardinality matching in general graphs.
   531 - \ref MaxWeightedMatching Edmond's blossom shrinking algorithm for calculating
   532   maximum weighted matching in general graphs.
   533 - \ref MaxWeightedPerfectMatching
   534   Edmond's blossom shrinking algorithm for calculating maximum weighted
   535   perfect matching in general graphs.
   536 
   537 \image html bipartite_matching.png
   538 \image latex bipartite_matching.eps "Bipartite Matching" width=\textwidth
   539 */
   540 
   541 /**
   542 @defgroup spantree Minimum Spanning Tree Algorithms
   543 @ingroup algs
   544 \brief Algorithms for finding a minimum cost spanning tree in a graph.
   545 
   546 This group contains the algorithms for finding a minimum cost spanning
   547 tree in a graph.
   548 */
   549 
   550 /**
   551 @defgroup auxalg Auxiliary Algorithms
   552 @ingroup algs
   553 \brief Auxiliary algorithms implemented in LEMON.
   554 
   555 This group contains some algorithms implemented in LEMON
   556 in order to make it easier to implement complex algorithms.
   557 */
   558 
   559 /**
   560 @defgroup approx Approximation Algorithms
   561 @ingroup algs
   562 \brief Approximation algorithms.
   563 
   564 This group contains the approximation and heuristic algorithms
   565 implemented in LEMON.
   566 */
   567 
   568 /**
   569 @defgroup gen_opt_group General Optimization Tools
   570 \brief This group contains some general optimization frameworks
   571 implemented in LEMON.
   572 
   573 This group contains some general optimization frameworks
   574 implemented in LEMON.
   575 */
   576 
   577 /**
   578 @defgroup lp_group Lp and Mip Solvers
   579 @ingroup gen_opt_group
   580 \brief Lp and Mip solver interfaces for LEMON.
   581 
   582 This group contains Lp and Mip solver interfaces for LEMON. The
   583 various LP solvers could be used in the same manner with this
   584 interface.
   585 */
   586 
   587 /**
   588 @defgroup lp_utils Tools for Lp and Mip Solvers
   589 @ingroup lp_group
   590 \brief Helper tools to the Lp and Mip solvers.
   591 
   592 This group adds some helper tools to general optimization framework
   593 implemented in LEMON.
   594 */
   595 
   596 /**
   597 @defgroup metah Metaheuristics
   598 @ingroup gen_opt_group
   599 \brief Metaheuristics for LEMON library.
   600 
   601 This group contains some metaheuristic optimization tools.
   602 */
   603 
   604 /**
   605 @defgroup utils Tools and Utilities
   606 \brief Tools and utilities for programming in LEMON
   607 
   608 Tools and utilities for programming in LEMON.
   609 */
   610 
   611 /**
   612 @defgroup gutils Basic Graph Utilities
   613 @ingroup utils
   614 \brief Simple basic graph utilities.
   615 
   616 This group contains some simple basic graph utilities.
   617 */
   618 
   619 /**
   620 @defgroup misc Miscellaneous Tools
   621 @ingroup utils
   622 \brief Tools for development, debugging and testing.
   623 
   624 This group contains several useful tools for development,
   625 debugging and testing.
   626 */
   627 
   628 /**
   629 @defgroup timecount Time Measuring and Counting
   630 @ingroup misc
   631 \brief Simple tools for measuring the performance of algorithms.
   632 
   633 This group contains simple tools for measuring the performance
   634 of algorithms.
   635 */
   636 
   637 /**
   638 @defgroup exceptions Exceptions
   639 @ingroup utils
   640 \brief Exceptions defined in LEMON.
   641 
   642 This group contains the exceptions defined in LEMON.
   643 */
   644 
   645 /**
   646 @defgroup io_group Input-Output
   647 \brief Graph Input-Output methods
   648 
   649 This group contains the tools for importing and exporting graphs
   650 and graph related data. Now it supports the \ref lgf-format
   651 "LEMON Graph Format", the \c DIMACS format and the encapsulated
   652 postscript (EPS) format.
   653 */
   654 
   655 /**
   656 @defgroup lemon_io LEMON Graph Format
   657 @ingroup io_group
   658 \brief Reading and writing LEMON Graph Format.
   659 
   660 This group contains methods for reading and writing
   661 \ref lgf-format "LEMON Graph Format".
   662 */
   663 
   664 /**
   665 @defgroup eps_io Postscript Exporting
   666 @ingroup io_group
   667 \brief General \c EPS drawer and graph exporter
   668 
   669 This group contains general \c EPS drawing methods and special
   670 graph exporting tools.
   671 */
   672 
   673 /**
   674 @defgroup dimacs_group DIMACS format
   675 @ingroup io_group
   676 \brief Read and write files in DIMACS format
   677 
   678 Tools to read a digraph from or write it to a file in DIMACS format data.
   679 */
   680 
   681 /**
   682 @defgroup nauty_group NAUTY Format
   683 @ingroup io_group
   684 \brief Read \e Nauty format
   685 
   686 Tool to read graphs from \e Nauty format data.
   687 */
   688 
   689 /**
   690 @defgroup concept Concepts
   691 \brief Skeleton classes and concept checking classes
   692 
   693 This group contains the data/algorithm skeletons and concept checking
   694 classes implemented in LEMON.
   695 
   696 The purpose of the classes in this group is fourfold.
   697 
   698 - These classes contain the documentations of the %concepts. In order
   699   to avoid document multiplications, an implementation of a concept
   700   simply refers to the corresponding concept class.
   701 
   702 - These classes declare every functions, <tt>typedef</tt>s etc. an
   703   implementation of the %concepts should provide, however completely
   704   without implementations and real data structures behind the
   705   interface. On the other hand they should provide nothing else. All
   706   the algorithms working on a data structure meeting a certain concept
   707   should compile with these classes. (Though it will not run properly,
   708   of course.) In this way it is easily to check if an algorithm
   709   doesn't use any extra feature of a certain implementation.
   710 
   711 - The concept descriptor classes also provide a <em>checker class</em>
   712   that makes it possible to check whether a certain implementation of a
   713   concept indeed provides all the required features.
   714 
   715 - Finally, They can serve as a skeleton of a new implementation of a concept.
   716 */
   717 
   718 /**
   719 @defgroup graph_concepts Graph Structure Concepts
   720 @ingroup concept
   721 \brief Skeleton and concept checking classes for graph structures
   722 
   723 This group contains the skeletons and concept checking classes of LEMON's
   724 graph structures and helper classes used to implement these.
   725 */
   726 
   727 /**
   728 @defgroup map_concepts Map Concepts
   729 @ingroup concept
   730 \brief Skeleton and concept checking classes for maps
   731 
   732 This group contains the skeletons and concept checking classes of maps.
   733 */
   734 
   735 /**
   736 \anchor demoprograms
   737 
   738 @defgroup demos Demo Programs
   739 
   740 Some demo programs are listed here. Their full source codes can be found in
   741 the \c demo subdirectory of the source tree.
   742 
   743 In order to compile them, use the <tt>make demo</tt> or the
   744 <tt>make check</tt> commands.
   745 */
   746 
   747 /**
   748 @defgroup tools Standalone Utility Applications
   749 
   750 Some utility applications are listed here.
   751 
   752 The standard compilation procedure (<tt>./configure;make</tt>) will compile
   753 them, as well.
   754 */
   755 
   756 }