COIN-OR::LEMON - Graph Library

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    33 * This file is a part of LEMON, a generic C++ optimization library.
    44 *
    5  * Copyright (C) 2003-2008
     5 * Copyright (C) 2003-2010
    66 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
    77 * (Egervary Research Group on Combinatorial Optimization, EGRES).
     
    1717 */
    1818
     19namespace lemon {
     20
    1921/**
    2022@defgroup datas Data Structures
    21 This group describes the several data structures implemented in LEMON.
     23This group contains the several data structures implemented in LEMON.
    2224*/
    2325
     
    6163
    6264/**
    63 @defgroup semi_adaptors Semi-Adaptor Classes for Graphs
     65@defgroup graph_adaptors Adaptor Classes for Graphs
    6466@ingroup graphs
    65 \brief Graph types between real graphs and graph adaptors.
    66 
    67 This group describes some graph types between real graphs and graph adaptors.
    68 These classes wrap graphs to give new functionality as the adaptors do it.
    69 On the other hand they are not light-weight structures as the adaptors.
     67\brief Adaptor classes for digraphs and graphs
     68
     69This group contains several useful adaptor classes for digraphs and graphs.
     70
     71The main parts of LEMON are the different graph structures, generic
     72graph algorithms, graph concepts, which couple them, and graph
     73adaptors. While the previous notions are more or less clear, the
     74latter one needs further explanation. Graph adaptors are graph classes
     75which serve for considering graph structures in different ways.
     76
     77A short example makes this much clearer.  Suppose that we have an
     78instance \c g of a directed graph type, say ListDigraph and an algorithm
     79\code
     80template <typename Digraph>
     81int algorithm(const Digraph&);
     82\endcode
     83is 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
     85arcs.  In this case, an adaptor class is used, which (according
     86to LEMON \ref concepts::Digraph "digraph concepts") works as a digraph.
     87The adaptor uses the original digraph structure and digraph operations when
     88methods of the reversed oriented graph are called.  This means that the adaptor
     89have minor memory usage, and do not perform sophisticated algorithmic
     90actions.  The purpose of it is to give a tool for the cases when a
     91graph have to be used in a specific alteration.  If this alteration is
     92obtained by a usual construction like filtering the node or the arc set or
     93considering a new orientation, then an adaptor is worthwhile to use.
     94To come back to the reverse oriented graph, in this situation
     95\code
     96template<typename Digraph> class ReverseDigraph;
     97\endcode
     98template class can be used. The code looks as follows
     99\code
     100ListDigraph g;
     101ReverseDigraph<ListDigraph> rg(g);
     102int result = algorithm(rg);
     103\endcode
     104During running the algorithm, the original digraph \c g is untouched.
     105This techniques give rise to an elegant code, and based on stable
     106graph adaptors, complex algorithms can be implemented easily.
     107
     108In flow, circulation and matching problems, the residual
     109graph is of particular importance. Combining an adaptor implementing
     110this with shortest path algorithms or minimum mean cycle algorithms,
     111a range of weighted and cardinality optimization algorithms can be
     112obtained. For other examples, the interested user is referred to the
     113detailed documentation of particular adaptors.
     114
     115The behavior of graph adaptors can be very different. Some of them keep
     116capabilities of the original graph while in other cases this would be
     117meaningless. This means that the concepts that they meet depend
     118on the graph adaptor, and the wrapped graph.
     119For example, if an arc of a reversed digraph is deleted, this is carried
     120out by deleting the corresponding arc of the original digraph, thus the
     121adaptor modifies the original digraph.
     122However in case of a residual digraph, this operation has no sense.
     123
     124Let us stand one more example here to simplify your work.
     125ReverseDigraph has constructor
     126\code
     127ReverseDigraph(Digraph& digraph);
     128\endcode
     129This means that in a situation, when a <tt>const %ListDigraph&</tt>
     130reference to a graph is given, then it have to be instantiated with
     131<tt>Digraph=const %ListDigraph</tt>.
     132\code
     133int algorithm1(const ListDigraph& g) {
     134  ReverseDigraph<const ListDigraph> rg(g);
     135  return algorithm2(rg);
     136}
     137\endcode
    70138*/
    71139
     
    75143\brief Map structures implemented in LEMON.
    76144
    77 This group describes the map structures implemented in LEMON.
     145This group contains the map structures implemented in LEMON.
    78146
    79147LEMON provides several special purpose maps and map adaptors that e.g. combine
     
    88156\brief Special graph-related maps.
    89157
    90 This group describes maps that are specifically designed to assign
    91 values to the nodes and arcs of graphs.
     158This group contains maps that are specifically designed to assign
     159values to the nodes and arcs/edges of graphs.
     160
     161If you are looking for the standard graph maps (\c NodeMap, \c ArcMap,
     162\c EdgeMap), see the \ref graph_concepts "Graph Structure Concepts".
    92163*/
    93164
     
    97168\brief Tools to create new maps from existing ones
    98169
    99 This group describes map adaptors that are used to create "implicit"
     170This group contains map adaptors that are used to create "implicit"
    100171maps from other maps.
    101172
    102 Most of them are \ref lemon::concepts::ReadMap "read-only maps".
     173Most of them are \ref concepts::ReadMap "read-only maps".
    103174They can make arithmetic and logical operations between one or two maps
    104175(negation, shifting, addition, multiplication, logical 'and', 'or',
     
    156227
    157228/**
    158 @defgroup matrices Matrices
    159 @ingroup datas
    160 \brief Two dimensional data storages implemented in LEMON.
    161 
    162 This group describes two dimensional data storages implemented in LEMON.
    163 */
    164 
    165 /**
    166229@defgroup paths Path Structures
    167230@ingroup datas
    168231\brief %Path structures implemented in LEMON.
    169232
    170 This group describes the path structures implemented in LEMON.
     233This group contains the path structures implemented in LEMON.
    171234
    172235LEMON provides flexible data structures to work with paths.
     
    176239any kind of path structure.
    177240
    178 \sa lemon::concepts::Path
     241\sa \ref concepts::Path "Path concept"
     242*/
     243
     244/**
     245@defgroup heaps Heap Structures
     246@ingroup datas
     247\brief %Heap structures implemented in LEMON.
     248
     249This group contains the heap structures implemented in LEMON.
     250
     251LEMON provides several heap classes. They are efficient implementations
     252of the abstract data type \e priority \e queue. They store items with
     253specified values called \e priorities in such a way that finding and
     254removing the item with minimum priority are efficient.
     255The basic operations are adding and erasing items, changing the priority
     256of an item, etc.
     257
     258Heaps are crucial in several algorithms, such as Dijkstra and Prim.
     259The heap implementations have the same interface, thus any of them can be
     260used easily in such algorithms.
     261
     262\sa \ref concepts::Heap "Heap concept"
    179263*/
    180264
     
    184268\brief Auxiliary data structures implemented in LEMON.
    185269
    186 This group describes some data structures implemented in LEMON in
     270This group contains some data structures implemented in LEMON in
    187271order to make it easier to implement combinatorial algorithms.
    188272*/
    189273
    190274/**
     275@defgroup geomdat Geometric Data Structures
     276@ingroup auxdat
     277\brief Geometric data structures implemented in LEMON.
     278
     279This group contains geometric data structures implemented in LEMON.
     280
     281 - \ref lemon::dim2::Point "dim2::Point" implements a two dimensional
     282   vector with the usual operations.
     283 - \ref lemon::dim2::Box "dim2::Box" can be used to determine the
     284   rectangular bounding box of a set of \ref lemon::dim2::Point
     285   "dim2::Point"'s.
     286*/
     287
     288/**
     289@defgroup matrices Matrices
     290@ingroup auxdat
     291\brief Two dimensional data storages implemented in LEMON.
     292
     293This group contains two dimensional data storages implemented in LEMON.
     294*/
     295
     296/**
    191297@defgroup algs Algorithms
    192 \brief This group describes the several algorithms
     298\brief This group contains the several algorithms
    193299implemented in LEMON.
    194300
    195 This group describes the several algorithms
     301This group contains the several algorithms
    196302implemented in LEMON.
    197303*/
     
    202308\brief Common graph search algorithms.
    203309
    204 This group describes the common graph search algorithms like
    205 Breadth-First Search (BFS) and Depth-First Search (DFS).
     310This group contains the common graph search algorithms, namely
     311\e breadth-first \e search (BFS) and \e depth-first \e search (DFS)
     312\ref clrs01algorithms.
    206313*/
    207314
     
    211318\brief Algorithms for finding shortest paths.
    212319
    213 This group describes the algorithms for finding shortest paths in graphs.
     320This group contains the algorithms for finding shortest paths in digraphs
     321\ref clrs01algorithms.
     322
     323 - \ref Dijkstra algorithm for finding shortest paths from a source node
     324   when all arc lengths are non-negative.
     325 - \ref BellmanFord "Bellman-Ford" algorithm for finding shortest paths
     326   from a source node when arc lenghts can be either positive or negative,
     327   but the digraph should not contain directed cycles with negative total
     328   length.
     329 - \ref FloydWarshall "Floyd-Warshall" and \ref Johnson "Johnson" algorithms
     330   for solving the \e all-pairs \e shortest \e paths \e problem when arc
     331   lenghts can be either positive or negative, but the digraph should
     332   not contain directed cycles with negative total length.
     333 - \ref Suurballe A successive shortest path algorithm for finding
     334   arc-disjoint paths between two nodes having minimum total length.
     335*/
     336
     337/**
     338@defgroup spantree Minimum Spanning Tree Algorithms
     339@ingroup algs
     340\brief Algorithms for finding minimum cost spanning trees and arborescences.
     341
     342This group contains the algorithms for finding minimum cost spanning
     343trees and arborescences \ref clrs01algorithms.
    214344*/
    215345
     
    219349\brief Algorithms for finding maximum flows.
    220350
    221 This group describes the algorithms for finding maximum flows and
    222 feasible circulations.
    223 
    224 The maximum flow problem is to find a flow between a single source and
    225 a single target that is maximum. Formally, there is a \f$G=(V,A)\f$
    226 directed graph, an \f$c_a:A\rightarrow\mathbf{R}^+_0\f$ capacity
    227 function and given \f$s, t \in V\f$ source and target node. The
    228 maximum flow is the \f$f_a\f$ solution of the next optimization problem:
    229 
    230 \f[ 0 \le f_a \le c_a \f]
    231 \f[ \sum_{v\in\delta^{-}(u)}f_{vu}=\sum_{v\in\delta^{+}(u)}f_{uv}
    232 \qquad \forall u \in V \setminus \{s,t\}\f]
    233 \f[ \max \sum_{v\in\delta^{+}(s)}f_{uv} - \sum_{v\in\delta^{-}(s)}f_{vu}\f]
     351This group contains the algorithms for finding maximum flows and
     352feasible circulations \ref clrs01algorithms, \ref amo93networkflows.
     353
     354The \e maximum \e flow \e problem is to find a flow of maximum value between
     355a single source and a single target. Formally, there is a \f$G=(V,A)\f$
     356digraph, a \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function and
     357\f$s, t \in V\f$ source and target nodes.
     358A maximum flow is an \f$f: A\rightarrow\mathbf{R}^+_0\f$ solution of the
     359following optimization problem.
     360
     361\f[ \max\sum_{sv\in A} f(sv) - \sum_{vs\in A} f(vs) \f]
     362\f[ \sum_{uv\in A} f(uv) = \sum_{vu\in A} f(vu)
     363    \quad \forall u\in V\setminus\{s,t\} \f]
     364\f[ 0 \leq f(uv) \leq cap(uv) \quad \forall uv\in A \f]
    234365
    235366LEMON contains several algorithms for solving maximum flow problems:
    236 - \ref lemon::EdmondsKarp "Edmonds-Karp"
    237 - \ref lemon::Preflow "Goldberg's Preflow algorithm"
    238 - \ref lemon::DinitzSleatorTarjan "Dinitz's blocking flow algorithm with dynamic trees"
    239 - \ref lemon::GoldbergTarjan "Preflow algorithm with dynamic trees"
    240 
    241 In most cases the \ref lemon::Preflow "Preflow" algorithm provides the
    242 fastest method to compute the maximum flow. All impelementations
    243 provides functions to query the minimum cut, which is the dual linear
    244 programming problem of the maximum flow.
    245 */
    246 
    247 /**
    248 @defgroup min_cost_flow Minimum Cost Flow Algorithms
     367- \ref EdmondsKarp Edmonds-Karp algorithm
     368  \ref edmondskarp72theoretical.
     369- \ref Preflow Goldberg-Tarjan's preflow push-relabel algorithm
     370  \ref goldberg88newapproach.
     371- \ref DinitzSleatorTarjan Dinitz's blocking flow algorithm with dynamic trees
     372  \ref dinic70algorithm, \ref sleator83dynamic.
     373- \ref GoldbergTarjan !Preflow push-relabel algorithm with dynamic trees
     374  \ref goldberg88newapproach, \ref sleator83dynamic.
     375
     376In most cases the \ref Preflow algorithm provides the
     377fastest method for computing a maximum flow. All implementations
     378also provide functions to query the minimum cut, which is the dual
     379problem of maximum flow.
     380
     381\ref Circulation is a preflow push-relabel algorithm implemented directly
     382for finding feasible circulations, which is a somewhat different problem,
     383but it is strongly related to maximum flow.
     384For more information, see \ref Circulation.
     385*/
     386
     387/**
     388@defgroup min_cost_flow_algs Minimum Cost Flow Algorithms
    249389@ingroup algs
    250390
    251391\brief Algorithms for finding minimum cost flows and circulations.
    252392
    253 This group describes the algorithms for finding minimum cost flows and
    254 circulations.
     393This group contains the algorithms for finding minimum cost flows and
     394circulations \ref amo93networkflows. For more information about this
     395problem and its dual solution, see \ref min_cost_flow
     396"Minimum Cost Flow Problem".
     397
     398LEMON contains several algorithms for this problem.
     399 - \ref NetworkSimplex Primal Network Simplex algorithm with various
     400   pivot strategies \ref dantzig63linearprog, \ref kellyoneill91netsimplex.
     401 - \ref CostScaling Cost Scaling algorithm based on push/augment and
     402   relabel operations \ref goldberg90approximation, \ref goldberg97efficient,
     403   \ref bunnagel98efficient.
     404 - \ref CapacityScaling Capacity Scaling algorithm based on the successive
     405   shortest path method \ref edmondskarp72theoretical.
     406 - \ref CycleCanceling Cycle-Canceling algorithms, two of which are
     407   strongly polynomial \ref klein67primal, \ref goldberg89cyclecanceling.
     408
     409In general, \ref NetworkSimplex and \ref CostScaling are the most efficient
     410implementations, but the other two algorithms could be faster in special cases.
     411For example, if the total supply and/or capacities are rather small,
     412\ref CapacityScaling is usually the fastest algorithm (without effective scaling).
    255413*/
    256414
     
    261419\brief Algorithms for finding minimum cut in graphs.
    262420
    263 This group describes the algorithms for finding minimum cut in graphs.
    264 
    265 The minimum cut problem is to find a non-empty and non-complete
    266 \f$X\f$ subset of the vertices with minimum overall capacity on
    267 outgoing arcs. Formally, there is \f$G=(V,A)\f$ directed graph, an
    268 \f$c_a:A\rightarrow\mathbf{R}^+_0\f$ capacity function. The minimum
     421This group contains the algorithms for finding minimum cut in graphs.
     422
     423The \e minimum \e cut \e problem is to find a non-empty and non-complete
     424\f$X\f$ subset of the nodes with minimum overall capacity on
     425outgoing arcs. Formally, there is a \f$G=(V,A)\f$ digraph, a
     426\f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function. The minimum
    269427cut is the \f$X\f$ solution of the next optimization problem:
    270428
    271429\f[ \min_{X \subset V, X\not\in \{\emptyset, V\}}
    272 \sum_{uv\in A, u\in X, v\not\in X}c_{uv}\f]
     430    \sum_{uv\in A: u\in X, v\not\in X}cap(uv) \f]
    273431
    274432LEMON contains several algorithms related to minimum cut problems:
    275433
    276 - \ref lemon::HaoOrlin "Hao-Orlin algorithm" to calculate minimum cut
    277   in directed graphs
    278 - \ref lemon::NagamochiIbaraki "Nagamochi-Ibaraki algorithm" to
    279   calculate minimum cut in undirected graphs
    280 - \ref lemon::GomoryHuTree "Gomory-Hu tree computation" to calculate all
    281   pairs minimum cut in undirected graphs
     434- \ref HaoOrlin "Hao-Orlin algorithm" for calculating minimum cut
     435  in directed graphs.
     436- \ref NagamochiIbaraki "Nagamochi-Ibaraki algorithm" for
     437  calculating minimum cut in undirected graphs.
     438- \ref GomoryHu "Gomory-Hu tree computation" for calculating
     439  all-pairs minimum cut in undirected graphs.
    282440
    283441If you want to find minimum cut just between two distinict nodes,
    284 please see the \ref max_flow "Maximum Flow page".
    285 */
    286 
    287 /**
    288 @defgroup graph_prop Connectivity and Other Graph Properties
    289 @ingroup algs
    290 \brief Algorithms for discovering the graph properties
    291 
    292 This group describes the algorithms for discovering the graph properties
    293 like connectivity, bipartiteness, euler property, simplicity etc.
    294 
    295 \image html edge_biconnected_components.png
    296 \image latex edge_biconnected_components.eps "bi-edge-connected components" width=\textwidth
    297 */
    298 
    299 /**
    300 @defgroup planar Planarity Embedding and Drawing
    301 @ingroup algs
    302 \brief Algorithms for planarity checking, embedding and drawing
    303 
    304 This group describes the algorithms for planarity checking,
    305 embedding and drawing.
    306 
    307 \image html planar.png
    308 \image latex planar.eps "Plane graph" width=\textwidth
     442see the \ref max_flow "maximum flow problem".
     443*/
     444
     445/**
     446@defgroup min_mean_cycle Minimum Mean Cycle Algorithms
     447@ingroup algs
     448\brief Algorithms for finding minimum mean cycles.
     449
     450This group contains the algorithms for finding minimum mean cycles
     451\ref clrs01algorithms, \ref amo93networkflows.
     452
     453The \e minimum \e mean \e cycle \e problem is to find a directed cycle
     454of minimum mean length (cost) in a digraph.
     455The mean length of a cycle is the average length of its arcs, i.e. the
     456ratio between the total length of the cycle and the number of arcs on it.
     457
     458This problem has an important connection to \e conservative \e length
     459\e functions, too. A length function on the arcs of a digraph is called
     460conservative if and only if there is no directed cycle of negative total
     461length. For an arbitrary length function, the negative of the minimum
     462cycle mean is the smallest \f$\epsilon\f$ value so that increasing the
     463arc lengths uniformly by \f$\epsilon\f$ results in a conservative length
     464function.
     465
     466LEMON contains three algorithms for solving the minimum mean cycle problem:
     467- \ref KarpMmc Karp's original algorithm \ref amo93networkflows,
     468  \ref dasdan98minmeancycle.
     469- \ref HartmannOrlinMmc Hartmann-Orlin's algorithm, which is an improved
     470  version of Karp's algorithm \ref dasdan98minmeancycle.
     471- \ref HowardMmc Howard's policy iteration algorithm
     472  \ref dasdan98minmeancycle.
     473
     474In practice, the \ref HowardMmc "Howard" algorithm turned out to be by far the
     475most efficient one, though the best known theoretical bound on its running
     476time is exponential.
     477Both \ref KarpMmc "Karp" and \ref HartmannOrlinMmc "Hartmann-Orlin" algorithms
     478run in time O(ne) and use space O(n<sup>2</sup>+e), but the latter one is
     479typically faster due to the applied early termination scheme.
    309480*/
    310481
     
    314485\brief Algorithms for finding matchings in graphs and bipartite graphs.
    315486
    316 This group contains algorithm objects and functions to calculate
     487This group contains the algorithms for calculating
    317488matchings in graphs and bipartite graphs. The general matching problem is
    318 finding a subset of the arcs which does not shares common endpoints.
     489finding a subset of the edges for which each node has at most one incident
     490edge.
    319491
    320492There are several different algorithms for calculate matchings in
    321493graphs.  The matching problems in bipartite graphs are generally
    322494easier than in general graphs. The goal of the matching optimization
    323 can be the finding maximum cardinality, maximum weight or minimum cost
     495can be finding maximum cardinality, maximum weight or minimum cost
    324496matching. The search can be constrained to find perfect or
    325497maximum cardinality matching.
    326498
    327 LEMON contains the next algorithms:
    328 - \ref lemon::MaxBipartiteMatching "MaxBipartiteMatching" Hopcroft-Karp
    329   augmenting path algorithm for calculate maximum cardinality matching in
    330   bipartite graphs
    331 - \ref lemon::PrBipartiteMatching "PrBipartiteMatching" Push-Relabel
    332   algorithm for calculate maximum cardinality matching in bipartite graphs
    333 - \ref lemon::MaxWeightedBipartiteMatching "MaxWeightedBipartiteMatching"
    334   Successive shortest path algorithm for calculate maximum weighted matching
    335   and maximum weighted bipartite matching in bipartite graph
    336 - \ref lemon::MinCostMaxBipartiteMatching "MinCostMaxBipartiteMatching"
    337   Successive shortest path algorithm for calculate minimum cost maximum
    338   matching in bipartite graph
    339 - \ref lemon::MaxMatching "MaxMatching" Edmond's blossom shrinking algorithm
    340   for calculate maximum cardinality matching in general graph
    341 - \ref lemon::MaxWeightedMatching "MaxWeightedMatching" Edmond's blossom
    342   shrinking algorithm for calculate maximum weighted matching in general
    343   graph
    344 - \ref lemon::MaxWeightedPerfectMatching "MaxWeightedPerfectMatching"
    345   Edmond's blossom shrinking algorithm for calculate maximum weighted
    346   perfect matching in general graph
    347 
    348 \image html bipartite_matching.png
    349 \image latex bipartite_matching.eps "Bipartite Matching" width=\textwidth
    350 */
    351 
    352 /**
    353 @defgroup spantree Minimum Spanning Tree Algorithms
    354 @ingroup algs
    355 \brief Algorithms for finding a minimum cost spanning tree in a graph.
    356 
    357 This group describes the algorithms for finding a minimum cost spanning
    358 tree in a graph
     499The matching algorithms implemented in LEMON:
     500- \ref MaxBipartiteMatching Hopcroft-Karp augmenting path algorithm
     501  for calculating maximum cardinality matching in bipartite graphs.
     502- \ref PrBipartiteMatching Push-relabel algorithm
     503  for calculating maximum cardinality matching in bipartite graphs.
     504- \ref MaxWeightedBipartiteMatching
     505  Successive shortest path algorithm for calculating maximum weighted
     506  matching and maximum weighted bipartite matching in bipartite graphs.
     507- \ref MinCostMaxBipartiteMatching
     508  Successive shortest path algorithm for calculating minimum cost maximum
     509  matching in bipartite graphs.
     510- \ref MaxMatching Edmond's blossom shrinking algorithm for calculating
     511  maximum cardinality matching in general graphs.
     512- \ref MaxWeightedMatching Edmond's blossom shrinking algorithm for calculating
     513  maximum weighted matching in general graphs.
     514- \ref MaxWeightedPerfectMatching
     515  Edmond's blossom shrinking algorithm for calculating maximum weighted
     516  perfect matching in general graphs.
     517- \ref MaxFractionalMatching Push-relabel algorithm for calculating
     518  maximum cardinality fractional matching in general graphs.
     519- \ref MaxWeightedFractionalMatching Augmenting path algorithm for calculating
     520  maximum weighted fractional matching in general graphs.
     521- \ref MaxWeightedPerfectFractionalMatching
     522  Augmenting path algorithm for calculating maximum weighted
     523  perfect fractional matching in general graphs.
     524
     525\image html matching.png
     526\image latex matching.eps "Min Cost Perfect Matching" width=\textwidth
     527*/
     528
     529/**
     530@defgroup graph_properties Connectivity and Other Graph Properties
     531@ingroup algs
     532\brief Algorithms for discovering the graph properties
     533
     534This group contains the algorithms for discovering the graph properties
     535like connectivity, bipartiteness, euler property, simplicity etc.
     536
     537\image html connected_components.png
     538\image latex connected_components.eps "Connected components" width=\textwidth
     539*/
     540
     541/**
     542@defgroup planar Planar Embedding and Drawing
     543@ingroup algs
     544\brief Algorithms for planarity checking, embedding and drawing
     545
     546This group contains the algorithms for planarity checking,
     547embedding and drawing.
     548
     549\image html planar.png
     550\image latex planar.eps "Plane graph" width=\textwidth
     551*/
     552
     553/**
     554@defgroup approx_algs Approximation Algorithms
     555@ingroup algs
     556\brief Approximation algorithms.
     557
     558This group contains the approximation and heuristic algorithms
     559implemented in LEMON.
     560
     561<b>Maximum Clique Problem</b>
     562  - \ref GrossoLocatelliPullanMc An efficient heuristic algorithm of
     563    Grosso, Locatelli, and Pullan.
    359564*/
    360565
     
    364569\brief Auxiliary algorithms implemented in LEMON.
    365570
    366 This group describes some algorithms implemented in LEMON
     571This group contains some algorithms implemented in LEMON
    367572in order to make it easier to implement complex algorithms.
    368573*/
    369574
    370575/**
    371 @defgroup approx Approximation Algorithms
    372 @ingroup algs
    373 \brief Approximation algorithms.
    374 
    375 This group describes the approximation and heuristic algorithms
     576@defgroup gen_opt_group General Optimization Tools
     577\brief This group contains some general optimization frameworks
    376578implemented in LEMON.
    377 */
    378 
    379 /**
    380 @defgroup gen_opt_group General Optimization Tools
    381 \brief This group describes some general optimization frameworks
     579
     580This group contains some general optimization frameworks
    382581implemented in LEMON.
    383 
    384 This group describes some general optimization frameworks
    385 implemented in LEMON.
    386 */
    387 
    388 /**
    389 @defgroup lp_group Lp and Mip Solvers
     582*/
     583
     584/**
     585@defgroup lp_group LP and MIP Solvers
    390586@ingroup gen_opt_group
    391 \brief Lp and Mip solver interfaces for LEMON.
    392 
    393 This group describes Lp and Mip solver interfaces for LEMON. The
    394 various LP solvers could be used in the same manner with this
    395 interface.
     587\brief LP and MIP solver interfaces for LEMON.
     588
     589This group contains LP and MIP solver interfaces for LEMON.
     590Various LP solvers could be used in the same manner with this
     591high-level interface.
     592
     593The currently supported solvers are \ref glpk, \ref clp, \ref cbc,
     594\ref cplex, \ref soplex.
    396595*/
    397596
     
    410609\brief Metaheuristics for LEMON library.
    411610
    412 This group describes some metaheuristic optimization tools.
     611This group contains some metaheuristic optimization tools.
    413612*/
    414613
     
    425624\brief Simple basic graph utilities.
    426625
    427 This group describes some simple basic graph utilities.
     626This group contains some simple basic graph utilities.
    428627*/
    429628
     
    433632\brief Tools for development, debugging and testing.
    434633
    435 This group describes several useful tools for development,
     634This group contains several useful tools for development,
    436635debugging and testing.
    437636*/
     
    442641\brief Simple tools for measuring the performance of algorithms.
    443642
    444 This group describes simple tools for measuring the performance
     643This group contains simple tools for measuring the performance
    445644of algorithms.
    446645*/
     
    451650\brief Exceptions defined in LEMON.
    452651
    453 This group describes the exceptions defined in LEMON.
     652This group contains the exceptions defined in LEMON.
    454653*/
    455654
     
    458657\brief Graph Input-Output methods
    459658
    460 This group describes the tools for importing and exporting graphs
     659This group contains the tools for importing and exporting graphs
    461660and graph related data. Now it supports the \ref lgf-format
    462661"LEMON Graph Format", the \c DIMACS format and the encapsulated
     
    465664
    466665/**
    467 @defgroup lemon_io LEMON Input-Output
     666@defgroup lemon_io LEMON Graph Format
    468667@ingroup io_group
    469668\brief Reading and writing LEMON Graph Format.
    470669
    471 This group describes methods for reading and writing
     670This group contains methods for reading and writing
    472671\ref lgf-format "LEMON Graph Format".
    473672*/
     
    478677\brief General \c EPS drawer and graph exporter
    479678
    480 This group describes general \c EPS drawing methods and special
     679This group contains general \c EPS drawing methods and special
    481680graph exporting tools.
     681*/
     682
     683/**
     684@defgroup dimacs_group DIMACS Format
     685@ingroup io_group
     686\brief Read and write files in DIMACS format
     687
     688Tools to read a digraph from or write it to a file in DIMACS format data.
     689*/
     690
     691/**
     692@defgroup nauty_group NAUTY Format
     693@ingroup io_group
     694\brief Read \e Nauty format
     695
     696Tool to read graphs from \e Nauty format data.
    482697*/
    483698
     
    486701\brief Skeleton classes and concept checking classes
    487702
    488 This group describes the data/algorithm skeletons and concept checking
     703This group contains the data/algorithm skeletons and concept checking
    489704classes implemented in LEMON.
    490705
     
    516731\brief Skeleton and concept checking classes for graph structures
    517732
    518 This group describes the skeletons and concept checking classes of LEMON's
    519 graph structures and helper classes used to implement these.
     733This group contains the skeletons and concept checking classes of
     734graph structures.
    520735*/
    521736
     
    525740\brief Skeleton and concept checking classes for maps
    526741
    527 This group describes the skeletons and concept checking classes of maps.
     742This group contains the skeletons and concept checking classes of maps.
     743*/
     744
     745/**
     746@defgroup tools Standalone Utility Applications
     747
     748Some utility applications are listed here.
     749
     750The standard compilation procedure (<tt>./configure;make</tt>) will compile
     751them, as well.
    528752*/
    529753
     
    531755\anchor demoprograms
    532756
    533 @defgroup demos Demo programs
     757@defgroup demos Demo Programs
    534758
    535759Some demo programs are listed here. Their full source codes can be found in
    536760the \c demo subdirectory of the source tree.
    537761
    538 It order to compile them, use <tt>--enable-demo</tt> configure option when
    539 build the library.
    540 */
    541 
    542 /**
    543 @defgroup tools Standalone utility applications
    544 
    545 Some utility applications are listed here.
    546 
    547 The standard compilation procedure (<tt>./configure;make</tt>) will compile
    548 them, as well.
    549 */
    550 
     762In order to compile them, use the <tt>make demo</tt> or the
     763<tt>make check</tt> commands.
     764*/
     765
     766}
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