doc/groups.dox
author Peter Kovacs <kpeter@inf.elte.hu>
Sat, 07 Oct 2017 15:48:00 +0200
changeset 1410 d9f79b81ef6c
parent 1351 2f479109a71d
permissions -rw-r--r--
Change the default graph type of Vf2 and Vf2pp (#597)
     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) 2003-2013
     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 Since the adaptor classes conform to the \ref graph_concepts "graph concepts",
   116 an adaptor can even be applied to another one.
   117 The following image illustrates a situation when a \ref SubDigraph adaptor
   118 is applied on a digraph and \ref Undirector is applied on the subgraph.
   119 
   120 \image html adaptors2.png
   121 \image latex adaptors2.eps "Using graph adaptors" width=\textwidth
   122 
   123 The behavior of graph adaptors can be very different. Some of them keep
   124 capabilities of the original graph while in other cases this would be
   125 meaningless. This means that the concepts that they meet depend
   126 on the graph adaptor, and the wrapped graph.
   127 For example, if an arc of a reversed digraph is deleted, this is carried
   128 out by deleting the corresponding arc of the original digraph, thus the
   129 adaptor modifies the original digraph.
   130 However in case of a residual digraph, this operation has no sense.
   131 
   132 Let us stand one more example here to simplify your work.
   133 ReverseDigraph has constructor
   134 \code
   135 ReverseDigraph(Digraph& digraph);
   136 \endcode
   137 This means that in a situation, when a <tt>const %ListDigraph&</tt>
   138 reference to a graph is given, then it have to be instantiated with
   139 <tt>Digraph=const %ListDigraph</tt>.
   140 \code
   141 int algorithm1(const ListDigraph& g) {
   142   ReverseDigraph<const ListDigraph> rg(g);
   143   return algorithm2(rg);
   144 }
   145 \endcode
   146 */
   147 
   148 /**
   149 @defgroup maps Maps
   150 @ingroup datas
   151 \brief Map structures implemented in LEMON.
   152 
   153 This group contains the map structures implemented in LEMON.
   154 
   155 LEMON provides several special purpose maps and map adaptors that e.g. combine
   156 new maps from existing ones.
   157 
   158 <b>See also:</b> \ref map_concepts "Map Concepts".
   159 */
   160 
   161 /**
   162 @defgroup graph_maps Graph Maps
   163 @ingroup maps
   164 \brief Special graph-related maps.
   165 
   166 This group contains maps that are specifically designed to assign
   167 values to the nodes and arcs/edges of graphs.
   168 
   169 If you are looking for the standard graph maps (\c NodeMap, \c ArcMap,
   170 \c EdgeMap), see the \ref graph_concepts "Graph Structure Concepts".
   171 */
   172 
   173 /**
   174 \defgroup map_adaptors Map Adaptors
   175 \ingroup maps
   176 \brief Tools to create new maps from existing ones
   177 
   178 This group contains map adaptors that are used to create "implicit"
   179 maps from other maps.
   180 
   181 Most of them are \ref concepts::ReadMap "read-only maps".
   182 They can make arithmetic and logical operations between one or two maps
   183 (negation, shifting, addition, multiplication, logical 'and', 'or',
   184 'not' etc.) or e.g. convert a map to another one of different Value type.
   185 
   186 The typical usage of this classes is passing implicit maps to
   187 algorithms.  If a function type algorithm is called then the function
   188 type map adaptors can be used comfortable. For example let's see the
   189 usage of map adaptors with the \c graphToEps() function.
   190 \code
   191   Color nodeColor(int deg) {
   192     if (deg >= 2) {
   193       return Color(0.5, 0.0, 0.5);
   194     } else if (deg == 1) {
   195       return Color(1.0, 0.5, 1.0);
   196     } else {
   197       return Color(0.0, 0.0, 0.0);
   198     }
   199   }
   200 
   201   Digraph::NodeMap<int> degree_map(graph);
   202 
   203   graphToEps(graph, "graph.eps")
   204     .coords(coords).scaleToA4().undirected()
   205     .nodeColors(composeMap(functorToMap(nodeColor), degree_map))
   206     .run();
   207 \endcode
   208 The \c functorToMap() function makes an \c int to \c Color map from the
   209 \c nodeColor() function. The \c composeMap() compose the \c degree_map
   210 and the previously created map. The composed map is a proper function to
   211 get the color of each node.
   212 
   213 The usage with class type algorithms is little bit harder. In this
   214 case the function type map adaptors can not be used, because the
   215 function map adaptors give back temporary objects.
   216 \code
   217   Digraph graph;
   218 
   219   typedef Digraph::ArcMap<double> DoubleArcMap;
   220   DoubleArcMap length(graph);
   221   DoubleArcMap speed(graph);
   222 
   223   typedef DivMap<DoubleArcMap, DoubleArcMap> TimeMap;
   224   TimeMap time(length, speed);
   225 
   226   Dijkstra<Digraph, TimeMap> dijkstra(graph, time);
   227   dijkstra.run(source, target);
   228 \endcode
   229 We have a length map and a maximum speed map on the arcs of a digraph.
   230 The minimum time to pass the arc can be calculated as the division of
   231 the two maps which can be done implicitly with the \c DivMap template
   232 class. We use the implicit minimum time map as the length map of the
   233 \c Dijkstra algorithm.
   234 */
   235 
   236 /**
   237 @defgroup paths Path Structures
   238 @ingroup datas
   239 \brief %Path structures implemented in LEMON.
   240 
   241 This group contains the path structures implemented in LEMON.
   242 
   243 LEMON provides flexible data structures to work with paths.
   244 All of them have similar interfaces and they can be copied easily with
   245 assignment operators and copy constructors. This makes it easy and
   246 efficient to have e.g. the Dijkstra algorithm to store its result in
   247 any kind of path structure.
   248 
   249 \sa \ref concepts::Path "Path concept"
   250 */
   251 
   252 /**
   253 @defgroup heaps Heap Structures
   254 @ingroup datas
   255 \brief %Heap structures implemented in LEMON.
   256 
   257 This group contains the heap structures implemented in LEMON.
   258 
   259 LEMON provides several heap classes. They are efficient implementations
   260 of the abstract data type \e priority \e queue. They store items with
   261 specified values called \e priorities in such a way that finding and
   262 removing the item with minimum priority are efficient.
   263 The basic operations are adding and erasing items, changing the priority
   264 of an item, etc.
   265 
   266 Heaps are crucial in several algorithms, such as Dijkstra and Prim.
   267 The heap implementations have the same interface, thus any of them can be
   268 used easily in such algorithms.
   269 
   270 \sa \ref concepts::Heap "Heap concept"
   271 */
   272 
   273 /**
   274 @defgroup auxdat Auxiliary Data Structures
   275 @ingroup datas
   276 \brief Auxiliary data structures implemented in LEMON.
   277 
   278 This group contains some data structures implemented in LEMON in
   279 order to make it easier to implement combinatorial algorithms.
   280 */
   281 
   282 /**
   283 @defgroup geomdat Geometric Data Structures
   284 @ingroup auxdat
   285 \brief Geometric data structures implemented in LEMON.
   286 
   287 This group contains geometric data structures implemented in LEMON.
   288 
   289  - \ref lemon::dim2::Point "dim2::Point" implements a two dimensional
   290    vector with the usual operations.
   291  - \ref lemon::dim2::Box "dim2::Box" can be used to determine the
   292    rectangular bounding box of a set of \ref lemon::dim2::Point
   293    "dim2::Point"'s.
   294 */
   295 
   296 /**
   297 @defgroup matrices Matrices
   298 @ingroup auxdat
   299 \brief Two dimensional data storages implemented in LEMON.
   300 
   301 This group contains two dimensional data storages implemented in LEMON.
   302 */
   303 
   304 /**
   305 @defgroup algs Algorithms
   306 \brief This group contains the several algorithms
   307 implemented in LEMON.
   308 
   309 This group contains the several algorithms
   310 implemented in LEMON.
   311 */
   312 
   313 /**
   314 @defgroup search Graph Search
   315 @ingroup algs
   316 \brief Common graph search algorithms.
   317 
   318 This group contains the common graph search algorithms, namely
   319 \e breadth-first \e search (BFS) and \e depth-first \e search (DFS)
   320 \cite clrs01algorithms.
   321 */
   322 
   323 /**
   324 @defgroup shortest_path Shortest Path Algorithms
   325 @ingroup algs
   326 \brief Algorithms for finding shortest paths.
   327 
   328 This group contains the algorithms for finding shortest paths in digraphs
   329 \cite clrs01algorithms.
   330 
   331  - \ref Dijkstra algorithm for finding shortest paths from a source node
   332    when all arc lengths are non-negative.
   333  - \ref BellmanFord "Bellman-Ford" algorithm for finding shortest paths
   334    from a source node when arc lenghts can be either positive or negative,
   335    but the digraph should not contain directed cycles with negative total
   336    length.
   337  - \ref FloydWarshall "Floyd-Warshall" and \ref Johnson "Johnson" algorithms
   338    for solving the \e all-pairs \e shortest \e paths \e problem when arc
   339    lenghts can be either positive or negative, but the digraph should
   340    not contain directed cycles with negative total length.
   341  - \ref Suurballe A successive shortest path algorithm for finding
   342    arc-disjoint paths between two nodes having minimum total length.
   343 */
   344 
   345 /**
   346 @defgroup spantree Minimum Spanning Tree Algorithms
   347 @ingroup algs
   348 \brief Algorithms for finding minimum cost spanning trees and arborescences.
   349 
   350 This group contains the algorithms for finding minimum cost spanning
   351 trees and arborescences \cite clrs01algorithms.
   352 */
   353 
   354 /**
   355 @defgroup max_flow Maximum Flow Algorithms
   356 @ingroup algs
   357 \brief Algorithms for finding maximum flows.
   358 
   359 This group contains the algorithms for finding maximum flows and
   360 feasible circulations \cite clrs01algorithms, \cite amo93networkflows.
   361 
   362 The \e maximum \e flow \e problem is to find a flow of maximum value between
   363 a single source and a single target. Formally, there is a \f$G=(V,A)\f$
   364 digraph, a \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function and
   365 \f$s, t \in V\f$ source and target nodes.
   366 A maximum flow is an \f$f: A\rightarrow\mathbf{R}^+_0\f$ solution of the
   367 following optimization problem.
   368 
   369 \f[ \max\sum_{sv\in A} f(sv) - \sum_{vs\in A} f(vs) \f]
   370 \f[ \sum_{uv\in A} f(uv) = \sum_{vu\in A} f(vu)
   371     \quad \forall u\in V\setminus\{s,t\} \f]
   372 \f[ 0 \leq f(uv) \leq cap(uv) \quad \forall uv\in A \f]
   373 
   374 LEMON contains several algorithms for solving maximum flow problems:
   375 - \ref EdmondsKarp Edmonds-Karp algorithm
   376   \cite edmondskarp72theoretical.
   377 - \ref Preflow Goldberg-Tarjan's preflow push-relabel algorithm
   378   \cite goldberg88newapproach.
   379 - \ref DinitzSleatorTarjan Dinitz's blocking flow algorithm with dynamic trees
   380   \cite dinic70algorithm, \cite sleator83dynamic.
   381 - \ref GoldbergTarjan !Preflow push-relabel algorithm with dynamic trees
   382   \cite goldberg88newapproach, \cite sleator83dynamic.
   383 
   384 In most cases the \ref Preflow algorithm provides the
   385 fastest method for computing a maximum flow. All implementations
   386 also provide functions to query the minimum cut, which is the dual
   387 problem of maximum flow.
   388 
   389 \ref Circulation is a preflow push-relabel algorithm implemented directly
   390 for finding feasible circulations, which is a somewhat different problem,
   391 but it is strongly related to maximum flow.
   392 For more information, see \ref Circulation.
   393 */
   394 
   395 /**
   396 @defgroup min_cost_flow_algs Minimum Cost Flow Algorithms
   397 @ingroup algs
   398 
   399 \brief Algorithms for finding minimum cost flows and circulations.
   400 
   401 This group contains the algorithms for finding minimum cost flows and
   402 circulations \cite amo93networkflows. For more information about this
   403 problem and its dual solution, see: \ref min_cost_flow
   404 "Minimum Cost Flow Problem".
   405 
   406 LEMON contains several algorithms for this problem.
   407  - \ref NetworkSimplex Primal Network Simplex algorithm with various
   408    pivot strategies \cite dantzig63linearprog, \cite kellyoneill91netsimplex.
   409  - \ref CostScaling Cost Scaling algorithm based on push/augment and
   410    relabel operations \cite goldberg90approximation, \cite goldberg97efficient,
   411    \cite bunnagel98efficient.
   412  - \ref CapacityScaling Capacity Scaling algorithm based on the successive
   413    shortest path method \cite edmondskarp72theoretical.
   414  - \ref CycleCanceling Cycle-Canceling algorithms, two of which are
   415    strongly polynomial \cite klein67primal, \cite goldberg89cyclecanceling.
   416 
   417 In general, \ref NetworkSimplex and \ref CostScaling are the most efficient
   418 implementations.
   419 \ref NetworkSimplex is usually the fastest on relatively small graphs (up to
   420 several thousands of nodes) and on dense graphs, while \ref CostScaling is
   421 typically more efficient on large graphs (e.g. hundreds of thousands of
   422 nodes or above), especially if they are sparse.
   423 However, other algorithms could be faster in special cases.
   424 For example, if the total supply and/or capacities are rather small,
   425 \ref CapacityScaling is usually the fastest algorithm
   426 (without effective scaling).
   427 
   428 These classes are intended to be used with integer-valued input data
   429 (capacities, supply values, and costs), except for \ref CapacityScaling,
   430 which is capable of handling real-valued arc costs (other numerical
   431 data are required to be integer).
   432 
   433 For more details about these implementations and for a comprehensive
   434 experimental study, see the paper \cite KiralyKovacs12MCF.
   435 It also compares these codes to other publicly available
   436 minimum cost flow solvers.
   437 */
   438 
   439 /**
   440 @defgroup min_cut Minimum Cut Algorithms
   441 @ingroup algs
   442 
   443 \brief Algorithms for finding minimum cut in graphs.
   444 
   445 This group contains the algorithms for finding minimum cut in graphs.
   446 
   447 The \e minimum \e cut \e problem is to find a non-empty and non-complete
   448 \f$X\f$ subset of the nodes with minimum overall capacity on
   449 outgoing arcs. Formally, there is a \f$G=(V,A)\f$ digraph, a
   450 \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function. The minimum
   451 cut is the \f$X\f$ solution of the next optimization problem:
   452 
   453 \f[ \min_{X \subset V, X\not\in \{\emptyset, V\}}
   454     \sum_{uv\in A: u\in X, v\not\in X}cap(uv) \f]
   455 
   456 LEMON contains several algorithms related to minimum cut problems:
   457 
   458 - \ref HaoOrlin "Hao-Orlin algorithm" for calculating minimum cut
   459   in directed graphs.
   460 - \ref NagamochiIbaraki "Nagamochi-Ibaraki algorithm" for
   461   calculating minimum cut in undirected graphs.
   462 - \ref GomoryHu "Gomory-Hu tree computation" for calculating
   463   all-pairs minimum cut in undirected graphs.
   464 
   465 If you want to find minimum cut just between two distinict nodes,
   466 see the \ref max_flow "maximum flow problem".
   467 */
   468 
   469 /**
   470 @defgroup min_mean_cycle Minimum Mean Cycle Algorithms
   471 @ingroup algs
   472 \brief Algorithms for finding minimum mean cycles.
   473 
   474 This group contains the algorithms for finding minimum mean cycles
   475 \cite amo93networkflows, \cite karp78characterization.
   476 
   477 The \e minimum \e mean \e cycle \e problem is to find a directed cycle
   478 of minimum mean length (cost) in a digraph.
   479 The mean length of a cycle is the average length of its arcs, i.e. the
   480 ratio between the total length of the cycle and the number of arcs on it.
   481 
   482 This problem has an important connection to \e conservative \e length
   483 \e functions, too. A length function on the arcs of a digraph is called
   484 conservative if and only if there is no directed cycle of negative total
   485 length. For an arbitrary length function, the negative of the minimum
   486 cycle mean is the smallest \f$\epsilon\f$ value so that increasing the
   487 arc lengths uniformly by \f$\epsilon\f$ results in a conservative length
   488 function.
   489 
   490 LEMON contains three algorithms for solving the minimum mean cycle problem:
   491 - \ref KarpMmc Karp's original algorithm \cite karp78characterization.
   492 - \ref HartmannOrlinMmc Hartmann-Orlin's algorithm, which is an improved
   493   version of Karp's algorithm \cite hartmann93finding.
   494 - \ref HowardMmc Howard's policy iteration algorithm
   495   \cite dasdan98minmeancycle, \cite dasdan04experimental.
   496 
   497 In practice, the \ref HowardMmc "Howard" algorithm turned out to be by far the
   498 most efficient one, though the best known theoretical bound on its running
   499 time is exponential.
   500 Both \ref KarpMmc "Karp" and \ref HartmannOrlinMmc "Hartmann-Orlin" algorithms
   501 run in time O(nm) and use space O(n<sup>2</sup>+m).
   502 */
   503 
   504 /**
   505 @defgroup matching Matching Algorithms
   506 @ingroup algs
   507 \brief Algorithms for finding matchings in graphs and bipartite graphs.
   508 
   509 This group contains the algorithms for calculating
   510 matchings in graphs and bipartite graphs. The general matching problem is
   511 finding a subset of the edges for which each node has at most one incident
   512 edge.
   513 
   514 There are several different algorithms for calculate matchings in
   515 graphs.  The matching problems in bipartite graphs are generally
   516 easier than in general graphs. The goal of the matching optimization
   517 can be finding maximum cardinality, maximum weight or minimum cost
   518 matching. The search can be constrained to find perfect or
   519 maximum cardinality matching.
   520 
   521 The matching algorithms implemented in LEMON:
   522 - \ref MaxBipartiteMatching Hopcroft-Karp augmenting path algorithm
   523   for calculating maximum cardinality matching in bipartite graphs.
   524 - \ref PrBipartiteMatching Push-relabel algorithm
   525   for calculating maximum cardinality matching in bipartite graphs.
   526 - \ref MaxWeightedBipartiteMatching
   527   Successive shortest path algorithm for calculating maximum weighted
   528   matching and maximum weighted bipartite matching in bipartite graphs.
   529 - \ref MinCostMaxBipartiteMatching
   530   Successive shortest path algorithm for calculating minimum cost maximum
   531   matching in bipartite graphs.
   532 - \ref MaxMatching Edmond's blossom shrinking algorithm for calculating
   533   maximum cardinality matching in general graphs.
   534 - \ref MaxWeightedMatching Edmond's blossom shrinking algorithm for calculating
   535   maximum weighted matching in general graphs.
   536 - \ref MaxWeightedPerfectMatching
   537   Edmond's blossom shrinking algorithm for calculating maximum weighted
   538   perfect matching in general graphs.
   539 - \ref MaxFractionalMatching Push-relabel algorithm for calculating
   540   maximum cardinality fractional matching in general graphs.
   541 - \ref MaxWeightedFractionalMatching Augmenting path algorithm for calculating
   542   maximum weighted fractional matching in general graphs.
   543 - \ref MaxWeightedPerfectFractionalMatching
   544   Augmenting path algorithm for calculating maximum weighted
   545   perfect fractional matching in general graphs.
   546 
   547 \image html matching.png
   548 \image latex matching.eps "Min Cost Perfect Matching" width=\textwidth
   549 */
   550 
   551 /**
   552 @defgroup graph_properties Connectivity and Other Graph Properties
   553 @ingroup algs
   554 \brief Algorithms for discovering the graph properties
   555 
   556 This group contains the algorithms for discovering the graph properties
   557 like connectivity, bipartiteness, euler property, simplicity etc.
   558 
   559 \image html connected_components.png
   560 \image latex connected_components.eps "Connected components" width=\textwidth
   561 */
   562 
   563 /**
   564 @defgroup graph_isomorphism Graph Isomorphism
   565 @ingroup algs
   566 \brief Algorithms for testing (sub)graph isomorphism
   567 
   568 This group contains algorithms for finding isomorph copies of a
   569 given graph in another one, or simply check whether two graphs are isomorphic.
   570 
   571 The formal definition of subgraph isomorphism is as follows.
   572 
   573 We are given two graphs, \f$G_1=(V_1,E_1)\f$ and \f$G_2=(V_2,E_2)\f$. A
   574 function \f$f:V_1\longrightarrow V_2\f$ is called \e mapping or \e
   575 embedding if \f$f(u)\neq f(v)\f$ whenever \f$u\neq v\f$.
   576 
   577 The standard <em>Subgraph Isomorphism Problem (SIP)</em> looks for a
   578 mapping with the property that whenever \f$(u,v)\in E_1\f$, then
   579 \f$(f(u),f(v))\in E_2\f$.
   580 
   581 In case of <em>Induced Subgraph Isomorphism Problem (ISIP)</em> one
   582 also requires that if \f$(u,v)\not\in E_1\f$, then \f$(f(u),f(v))\not\in
   583 E_2\f$
   584 
   585 In addition, the graph nodes may be \e labeled, i.e. we are given two
   586 node labelings \f$l_1:V_1\longrightarrow L\f$ and \f$l_2:V_2\longrightarrow
   587 L\f$ and we require that \f$l_1(u)=l_2(f(u))\f$ holds for all nodes \f$u \in
   588 G_1\f$.
   589 
   590 */
   591 
   592 /**
   593 @defgroup planar Planar Embedding and Drawing
   594 @ingroup algs
   595 \brief Algorithms for planarity checking, embedding and drawing
   596 
   597 This group contains the algorithms for planarity checking,
   598 embedding and drawing.
   599 
   600 \image html planar.png
   601 \image latex planar.eps "Plane graph" width=\textwidth
   602 */
   603 
   604 /**
   605 @defgroup tsp Traveling Salesman Problem
   606 @ingroup algs
   607 \brief Algorithms for the symmetric traveling salesman problem
   608 
   609 This group contains basic heuristic algorithms for the the symmetric
   610 \e traveling \e salesman \e problem (TSP).
   611 Given an \ref FullGraph "undirected full graph" with a cost map on its edges,
   612 the problem is to find a shortest possible tour that visits each node exactly
   613 once (i.e. the minimum cost Hamiltonian cycle).
   614 
   615 These TSP algorithms are intended to be used with a \e metric \e cost
   616 \e function, i.e. the edge costs should satisfy the triangle inequality.
   617 Otherwise the algorithms could yield worse results.
   618 
   619 LEMON provides five well-known heuristics for solving symmetric TSP:
   620  - \ref NearestNeighborTsp Neareast neighbor algorithm
   621  - \ref GreedyTsp Greedy algorithm
   622  - \ref InsertionTsp Insertion heuristic (with four selection methods)
   623  - \ref ChristofidesTsp Christofides algorithm
   624  - \ref Opt2Tsp 2-opt algorithm
   625 
   626 \ref NearestNeighborTsp, \ref GreedyTsp, and \ref InsertionTsp are the fastest
   627 solution methods. Furthermore, \ref InsertionTsp is usually quite effective.
   628 
   629 \ref ChristofidesTsp is somewhat slower, but it has the best guaranteed
   630 approximation factor: 3/2.
   631 
   632 \ref Opt2Tsp usually provides the best results in practice, but
   633 it is the slowest method. It can also be used to improve given tours,
   634 for example, the results of other algorithms.
   635 
   636 \image html tsp.png
   637 \image latex tsp.eps "Traveling salesman problem" width=\textwidth
   638 */
   639 
   640 /**
   641 @defgroup approx_algs Approximation Algorithms
   642 @ingroup algs
   643 \brief Approximation algorithms.
   644 
   645 This group contains the approximation and heuristic algorithms
   646 implemented in LEMON.
   647 
   648 <b>Maximum Clique Problem</b>
   649   - \ref GrossoLocatelliPullanMc An efficient heuristic algorithm of
   650     Grosso, Locatelli, and Pullan.
   651 */
   652 
   653 /**
   654 @defgroup auxalg Auxiliary Algorithms
   655 @ingroup algs
   656 \brief Auxiliary algorithms implemented in LEMON.
   657 
   658 This group contains some algorithms implemented in LEMON
   659 in order to make it easier to implement complex algorithms.
   660 */
   661 
   662 /**
   663 @defgroup gen_opt_group General Optimization Tools
   664 \brief This group contains some general optimization frameworks
   665 implemented in LEMON.
   666 
   667 This group contains some general optimization frameworks
   668 implemented in LEMON.
   669 */
   670 
   671 /**
   672 @defgroup lp_group LP and MIP Solvers
   673 @ingroup gen_opt_group
   674 \brief LP and MIP solver interfaces for LEMON.
   675 
   676 This group contains LP and MIP solver interfaces for LEMON.
   677 Various LP solvers could be used in the same manner with this
   678 high-level interface.
   679 
   680 The currently supported solvers are \cite glpk, \cite clp, \cite cbc,
   681 \cite cplex, \cite soplex.
   682 */
   683 
   684 /**
   685 @defgroup lp_utils Tools for Lp and Mip Solvers
   686 @ingroup lp_group
   687 \brief Helper tools to the Lp and Mip solvers.
   688 
   689 This group adds some helper tools to general optimization framework
   690 implemented in LEMON.
   691 */
   692 
   693 /**
   694 @defgroup metah Metaheuristics
   695 @ingroup gen_opt_group
   696 \brief Metaheuristics for LEMON library.
   697 
   698 This group contains some metaheuristic optimization tools.
   699 */
   700 
   701 /**
   702 @defgroup utils Tools and Utilities
   703 \brief Tools and utilities for programming in LEMON
   704 
   705 Tools and utilities for programming in LEMON.
   706 */
   707 
   708 /**
   709 @defgroup gutils Basic Graph Utilities
   710 @ingroup utils
   711 \brief Simple basic graph utilities.
   712 
   713 This group contains some simple basic graph utilities.
   714 */
   715 
   716 /**
   717 @defgroup misc Miscellaneous Tools
   718 @ingroup utils
   719 \brief Tools for development, debugging and testing.
   720 
   721 This group contains several useful tools for development,
   722 debugging and testing.
   723 */
   724 
   725 /**
   726 @defgroup timecount Time Measuring and Counting
   727 @ingroup misc
   728 \brief Simple tools for measuring the performance of algorithms.
   729 
   730 This group contains simple tools for measuring the performance
   731 of algorithms.
   732 */
   733 
   734 /**
   735 @defgroup exceptions Exceptions
   736 @ingroup utils
   737 \brief Exceptions defined in LEMON.
   738 
   739 This group contains the exceptions defined in LEMON.
   740 */
   741 
   742 /**
   743 @defgroup io_group Input-Output
   744 \brief Graph Input-Output methods
   745 
   746 This group contains the tools for importing and exporting graphs
   747 and graph related data. Now it supports the \ref lgf-format
   748 "LEMON Graph Format", the \c DIMACS format and the encapsulated
   749 postscript (EPS) format.
   750 */
   751 
   752 /**
   753 @defgroup lemon_io LEMON Graph Format
   754 @ingroup io_group
   755 \brief Reading and writing LEMON Graph Format.
   756 
   757 This group contains methods for reading and writing
   758 \ref lgf-format "LEMON Graph Format".
   759 */
   760 
   761 /**
   762 @defgroup eps_io Postscript Exporting
   763 @ingroup io_group
   764 \brief General \c EPS drawer and graph exporter
   765 
   766 This group contains general \c EPS drawing methods and special
   767 graph exporting tools.
   768 
   769 \image html graph_to_eps.png
   770 */
   771 
   772 /**
   773 @defgroup dimacs_group DIMACS Format
   774 @ingroup io_group
   775 \brief Read and write files in DIMACS format
   776 
   777 Tools to read a digraph from or write it to a file in DIMACS format data.
   778 */
   779 
   780 /**
   781 @defgroup nauty_group NAUTY Format
   782 @ingroup io_group
   783 \brief Read \e Nauty format
   784 
   785 Tool to read graphs from \e Nauty format data.
   786 */
   787 
   788 /**
   789 @defgroup concept Concepts
   790 \brief Skeleton classes and concept checking classes
   791 
   792 This group contains the data/algorithm skeletons and concept checking
   793 classes implemented in LEMON.
   794 
   795 The purpose of the classes in this group is fourfold.
   796 
   797 - These classes contain the documentations of the %concepts. In order
   798   to avoid document multiplications, an implementation of a concept
   799   simply refers to the corresponding concept class.
   800 
   801 - These classes declare every functions, <tt>typedef</tt>s etc. an
   802   implementation of the %concepts should provide, however completely
   803   without implementations and real data structures behind the
   804   interface. On the other hand they should provide nothing else. All
   805   the algorithms working on a data structure meeting a certain concept
   806   should compile with these classes. (Though it will not run properly,
   807   of course.) In this way it is easily to check if an algorithm
   808   doesn't use any extra feature of a certain implementation.
   809 
   810 - The concept descriptor classes also provide a <em>checker class</em>
   811   that makes it possible to check whether a certain implementation of a
   812   concept indeed provides all the required features.
   813 
   814 - Finally, They can serve as a skeleton of a new implementation of a concept.
   815 */
   816 
   817 /**
   818 @defgroup graph_concepts Graph Structure Concepts
   819 @ingroup concept
   820 \brief Skeleton and concept checking classes for graph structures
   821 
   822 This group contains the skeletons and concept checking classes of
   823 graph structures.
   824 */
   825 
   826 /**
   827 @defgroup map_concepts Map Concepts
   828 @ingroup concept
   829 \brief Skeleton and concept checking classes for maps
   830 
   831 This group contains the skeletons and concept checking classes of maps.
   832 */
   833 
   834 /**
   835 @defgroup tools Standalone Utility Applications
   836 
   837 Some utility applications are listed here.
   838 
   839 The standard compilation procedure (<tt>./configure;make</tt>) will compile
   840 them, as well.
   841 */
   842 
   843 /**
   844 \anchor demoprograms
   845 
   846 @defgroup demos Demo Programs
   847 
   848 Some demo programs are listed here. Their full source codes can be found in
   849 the \c demo subdirectory of the source tree.
   850 
   851 In order to compile them, use the <tt>make demo</tt> or the
   852 <tt>make check</tt> commands.
   853 */
   854 
   855 }