COIN-OR::LEMON - Graph Library

source: lemon/doc/groups.dox @ 687:6c408d864fa1

Last change on this file since 687:6c408d864fa1 was 687:6c408d864fa1, checked in by Peter Kovacs <kpeter@…>, 11 years ago

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.
File size: 26.2 KB
RevLine 
[209]1/* -*- mode: C++; indent-tabs-mode: nil; -*-
[40]2 *
[209]3 * This file is a part of LEMON, a generic C++ optimization library.
[40]4 *
[463]5 * Copyright (C) 2003-2009
[40]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
[422]19namespace lemon {
20
[40]21/**
22@defgroup datas Data Structures
[606]23This group contains the several data structures implemented in LEMON.
[40]24*/
25
26/**
27@defgroup graphs Graph Structures
28@ingroup datas
29\brief Graph structures implemented in LEMON.
30
[209]31The implementation of combinatorial algorithms heavily relies on
32efficient graph implementations. LEMON offers data structures which are
33planned to be easily used in an experimental phase of implementation studies,
34and thereafter the program code can be made efficient by small modifications.
[40]35
36The most efficient implementation of diverse applications require the
37usage of different physical graph implementations. These differences
38appear in the size of graph we require to handle, memory or time usage
39limitations or in the set of operations through which the graph can be
40accessed.  LEMON provides several physical graph structures to meet
41the diverging requirements of the possible users.  In order to save on
42running time or on memory usage, some structures may fail to provide
[83]43some graph features like arc/edge or node deletion.
[40]44
[209]45Alteration of standard containers need a very limited number of
46operations, these together satisfy the everyday requirements.
47In the case of graph structures, different operations are needed which do
48not alter the physical graph, but gives another view. If some nodes or
[83]49arcs have to be hidden or the reverse oriented graph have to be used, then
[209]50this is the case. It also may happen that in a flow implementation
51the residual graph can be accessed by another algorithm, or a node-set
52is to be shrunk for another algorithm.
53LEMON also provides a variety of graphs for these requirements called
54\ref graph_adaptors "graph adaptors". Adaptors cannot be used alone but only
55in conjunction with other graph representations.
[40]56
57You are free to use the graph structure that fit your requirements
58the best, most graph algorithms and auxiliary data structures can be used
[314]59with any graph structure.
60
61<b>See also:</b> \ref graph_concepts "Graph Structure Concepts".
[40]62*/
63
64/**
[474]65@defgroup graph_adaptors Adaptor Classes for Graphs
[432]66@ingroup graphs
[474]67\brief Adaptor classes for digraphs and graphs
68
69This group contains several useful adaptor classes for digraphs and graphs.
[432]70
71The main parts of LEMON are the different graph structures, generic
[474]72graph algorithms, graph concepts, which couple them, and graph
[432]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
[474]78instance \c g of a directed graph type, say ListDigraph and an algorithm
[432]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
[474]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
[432]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
[474]92obtained by a usual construction like filtering the node or the arc set or
[432]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;
[474]101ReverseDigraph<ListDigraph> rg(g);
[432]102int result = algorithm(rg);
103\endcode
[474]104During running the algorithm, the original digraph \c g is untouched.
105This techniques give rise to an elegant code, and based on stable
[432]106graph adaptors, complex algorithms can be implemented easily.
107
[474]108In flow, circulation and matching problems, the residual
[432]109graph is of particular importance. Combining an adaptor implementing
[474]110this with shortest path algorithms or minimum mean cycle algorithms,
[432]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
[474]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.
[432]123
124Let us stand one more example here to simplify your work.
[474]125ReverseDigraph has constructor
[432]126\code
127ReverseDigraph(Digraph& digraph);
128\endcode
[474]129This means that in a situation, when a <tt>const %ListDigraph&</tt>
[432]130reference to a graph is given, then it have to be instantiated with
[474]131<tt>Digraph=const %ListDigraph</tt>.
[432]132\code
133int algorithm1(const ListDigraph& g) {
[474]134  ReverseDigraph<const ListDigraph> rg(g);
[432]135  return algorithm2(rg);
136}
137\endcode
138*/
139
140/**
[50]141@defgroup semi_adaptors Semi-Adaptor Classes for Graphs
[40]142@ingroup graphs
143\brief Graph types between real graphs and graph adaptors.
144
[606]145This group contains some graph types between real graphs and graph adaptors.
[209]146These classes wrap graphs to give new functionality as the adaptors do it.
[50]147On the other hand they are not light-weight structures as the adaptors.
[40]148*/
149
150/**
[209]151@defgroup maps Maps
[40]152@ingroup datas
[50]153\brief Map structures implemented in LEMON.
[40]154
[606]155This group contains the map structures implemented in LEMON.
[50]156
[314]157LEMON provides several special purpose maps and map adaptors that e.g. combine
[40]158new maps from existing ones.
[314]159
160<b>See also:</b> \ref map_concepts "Map Concepts".
[40]161*/
162
163/**
[209]164@defgroup graph_maps Graph Maps
[40]165@ingroup maps
[83]166\brief Special graph-related maps.
[40]167
[606]168This group contains maps that are specifically designed to assign
[422]169values to the nodes and arcs/edges of graphs.
170
171If you are looking for the standard graph maps (\c NodeMap, \c ArcMap,
172\c EdgeMap), see the \ref graph_concepts "Graph Structure Concepts".
[40]173*/
174
175/**
176\defgroup map_adaptors Map Adaptors
177\ingroup maps
178\brief Tools to create new maps from existing ones
179
[606]180This group contains map adaptors that are used to create "implicit"
[50]181maps from other maps.
[40]182
[422]183Most of them are \ref concepts::ReadMap "read-only maps".
[83]184They 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.
[40]187
[50]188The typical usage of this classes is passing implicit maps to
[40]189algorithms.  If a function type algorithm is called then the function
190type map adaptors can be used comfortable. For example let's see the
[314]191usage of map adaptors with the \c graphToEps() function.
[40]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  }
[209]202
[83]203  Digraph::NodeMap<int> degree_map(graph);
[209]204
[314]205  graphToEps(graph, "graph.eps")
[40]206    .coords(coords).scaleToA4().undirected()
[83]207    .nodeColors(composeMap(functorToMap(nodeColor), degree_map))
[40]208    .run();
[209]209\endcode
[83]210The \c functorToMap() function makes an \c int to \c Color map from the
[314]211\c nodeColor() function. The \c composeMap() compose the \c degree_map
[83]212and the previously created map. The composed map is a proper function to
213get the color of each node.
[40]214
215The usage with class type algorithms is little bit harder. In this
216case the function type map adaptors can not be used, because the
[50]217function map adaptors give back temporary objects.
[40]218\code
[83]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;
[40]226  TimeMap time(length, speed);
[209]227
[83]228  Dijkstra<Digraph, TimeMap> dijkstra(graph, time);
[40]229  dijkstra.run(source, target);
230\endcode
[83]231We have a length map and a maximum speed map on the arcs of a digraph.
232The minimum time to pass the arc can be calculated as the division of
233the two maps which can be done implicitly with the \c DivMap template
[40]234class. We use the implicit minimum time map as the length map of the
235\c Dijkstra algorithm.
236*/
237
238/**
[209]239@defgroup matrices Matrices
[40]240@ingroup datas
[50]241\brief Two dimensional data storages implemented in LEMON.
[40]242
[606]243This group contains two dimensional data storages implemented in LEMON.
[40]244*/
245
246/**
247@defgroup paths Path Structures
248@ingroup datas
[318]249\brief %Path structures implemented in LEMON.
[40]250
[606]251This group contains the path structures implemented in LEMON.
[40]252
[50]253LEMON provides flexible data structures to work with paths.
254All of them have similar interfaces and they can be copied easily with
255assignment operators and copy constructors. This makes it easy and
[40]256efficient to have e.g. the Dijkstra algorithm to store its result in
257any kind of path structure.
258
259\sa lemon::concepts::Path
260*/
261
262/**
263@defgroup auxdat Auxiliary Data Structures
264@ingroup datas
[50]265\brief Auxiliary data structures implemented in LEMON.
[40]266
[606]267This group contains some data structures implemented in LEMON in
[40]268order to make it easier to implement combinatorial algorithms.
269*/
270
271/**
272@defgroup algs Algorithms
[606]273\brief This group contains the several algorithms
[40]274implemented in LEMON.
275
[606]276This group contains the several algorithms
[40]277implemented in LEMON.
278*/
279
280/**
281@defgroup search Graph Search
282@ingroup algs
[50]283\brief Common graph search algorithms.
[40]284
[606]285This group contains the common graph search algorithms, namely
[422]286\e breadth-first \e search (BFS) and \e depth-first \e search (DFS).
[40]287*/
288
289/**
[314]290@defgroup shortest_path Shortest Path Algorithms
[40]291@ingroup algs
[50]292\brief Algorithms for finding shortest paths.
[40]293
[606]294This group contains the algorithms for finding shortest paths in digraphs.
[422]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.
[40]308*/
309
[209]310/**
[314]311@defgroup max_flow Maximum Flow Algorithms
[209]312@ingroup algs
[50]313\brief Algorithms for finding maximum flows.
[40]314
[606]315This group contains the algorithms for finding maximum flows and
[40]316feasible circulations.
317
[422]318The \e maximum \e flow \e problem is to find a flow of maximum value between
319a single source and a single target. Formally, there is a \f$G=(V,A)\f$
[656]320digraph, a \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function and
[422]321\f$s, t \in V\f$ source and target nodes.
[656]322A maximum flow is an \f$f: A\rightarrow\mathbf{R}^+_0\f$ solution of the
[422]323following optimization problem.
[40]324
[656]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]
[40]329
[50]330LEMON contains several algorithms for solving maximum flow problems:
[422]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.
[40]335
[422]336In most cases the \ref Preflow "Preflow" algorithm provides the
337fastest method for computing a maximum flow. All implementations
338provides functions to also query the minimum cut, which is the dual
339problem of the maximum flow.
[40]340*/
341
342/**
[314]343@defgroup min_cost_flow Minimum Cost Flow Algorithms
[40]344@ingroup algs
345
[50]346\brief Algorithms for finding minimum cost flows and circulations.
[40]347
[656]348This group contains the algorithms for finding minimum cost flows and
[209]349circulations.
[422]350
351The \e minimum \e cost \e flow \e problem is to find a feasible flow of
352minimum total cost from a set of supply nodes to a set of demand nodes
[656]353in a network with capacity constraints (lower and upper bounds)
354and arc costs.
[687]355Formally, 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
[656]357upper bounds for the flow values on the arcs, for which
[687]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
360on the arcs and \f$sup: V\rightarrow\mathbf{Z}\f$ denotes the
[656]361signed supply values of the nodes.
362If \f$sup(u)>0\f$, then \f$u\f$ is a supply node with \f$sup(u)\f$
363supply, if \f$sup(u)<0\f$, then \f$u\f$ is a demand node with
364\f$-sup(u)\f$ demand.
[687]365A minimum cost flow is an \f$f: A\rightarrow\mathbf{Z}\f$ solution
[656]366of the following optimization problem.
[422]367
[656]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]
[422]372
[656]373The sum of the supply values, i.e. \f$\sum_{u\in V} sup(u)\f$ must be
374zero or negative in order to have a feasible solution (since the sum
375of the expressions on the left-hand side of the inequalities is zero).
376It means that the total demand must be greater or equal to the total
377supply and all the supplies have to be carried out from the supply nodes,
378but there could be demands that are not satisfied.
379If \f$\sum_{u\in V} sup(u)\f$ is zero, then all the supply/demand
380constraints have to be satisfied with equality, i.e. all demands
381have to be satisfied and all supplies have to be used.
382
383If 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
385have to be satisfied while there could be supplies that are not used),
386then you could easily transform the problem to the above form by reversing
387the direction of the arcs and taking the negative of the supply values
388(e.g. using \ref ReverseDigraph and \ref NegMap adaptors).
389However \ref NetworkSimplex algorithm also supports this form directly
390for the sake of convenience.
391
392A feasible solution for this problem can be found using \ref Circulation.
393
394Note that the above formulation is actually more general than the usual
395definition of the minimum cost flow problem, in which strict equalities
396are 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
401However if the sum of the supply values is zero, then these two problems
402are equivalent. So if you need the equality form, you have to ensure this
403additional contraint for the algorithms.
404
405The dual solution of the minimum cost flow problem is represented by node
406potentials \f$\pi: V\rightarrow\mathbf{Z}\f$.
[687]407An \f$f: A\rightarrow\mathbf{Z}\f$ feasible solution of the problem
[656]408is optimal if and only if for some \f$\pi: V\rightarrow\mathbf{Z}\f$
409node potentials the following \e complementary \e slackness optimality
410conditions 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$f(uv)=upper(uv)\f$.
[687]416 - For all \f$u\in V\f$ nodes:
[656]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 
420Here \f$cost^\pi(uv)\f$ denotes the \e reduced \e cost of the arc
[687]421\f$uv\in A\f$ with respect to the potential function \f$\pi\f$, i.e.
[656]422\f[ cost^\pi(uv) = cost(uv) + \pi(u) - \pi(v).\f]
423
[687]424All algorithms provide dual solution (node potentials) as well,
[656]425if an optimal flow is found.
426
427LEMON 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
[422]433   capacity scaling.
[656]434 - \ref CancelAndTighten The Cancel and Tighten algorithm.
435 - \ref CycleCanceling Cycle-Canceling algorithms.
436
437Most of these implementations support the general inequality form of the
438minimum cost flow problem, but CancelAndTighten and CycleCanceling
439only support the equality form due to the primal method they use.
440
441In general NetworkSimplex is the most efficient implementation,
442but in special cases other algorithms could be faster.
443For example, if the total supply and/or capacities are rather small,
444CapacityScaling is usually the fastest algorithm (without effective scaling).
[40]445*/
446
447/**
[314]448@defgroup min_cut Minimum Cut Algorithms
[209]449@ingroup algs
[40]450
[50]451\brief Algorithms for finding minimum cut in graphs.
[40]452
[606]453This group contains the algorithms for finding minimum cut in graphs.
[40]454
[422]455The \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
457outgoing 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
[50]459cut is the \f$X\f$ solution of the next optimization problem:
[40]460
[210]461\f[ \min_{X \subset V, X\not\in \{\emptyset, V\}}
[422]462    \sum_{uv\in A, u\in X, v\not\in X}cap(uv) \f]
[40]463
[50]464LEMON contains several algorithms related to minimum cut problems:
[40]465
[422]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.
[606]470- \ref GomoryHu "Gomory-Hu tree computation" for calculating
[422]471  all-pairs minimum cut in undirected graphs.
[40]472
473If you want to find minimum cut just between two distinict nodes,
[422]474see the \ref max_flow "maximum flow problem".
[40]475*/
476
477/**
[633]478@defgroup graph_properties Connectivity and Other Graph Properties
[40]479@ingroup algs
[50]480\brief Algorithms for discovering the graph properties
[40]481
[606]482This group contains the algorithms for discovering the graph properties
[50]483like connectivity, bipartiteness, euler property, simplicity etc.
[40]484
485\image html edge_biconnected_components.png
486\image latex edge_biconnected_components.eps "bi-edge-connected components" width=\textwidth
487*/
488
489/**
[314]490@defgroup planar Planarity Embedding and Drawing
[40]491@ingroup algs
[50]492\brief Algorithms for planarity checking, embedding and drawing
[40]493
[606]494This group contains the algorithms for planarity checking,
[210]495embedding and drawing.
[40]496
497\image html planar.png
498\image latex planar.eps "Plane graph" width=\textwidth
499*/
500
501/**
[314]502@defgroup matching Matching Algorithms
[40]503@ingroup algs
[50]504\brief Algorithms for finding matchings in graphs and bipartite graphs.
[40]505
[637]506This group contains the algorithms for calculating
[40]507matchings in graphs and bipartite graphs. The general matching problem is
[637]508finding a subset of the edges for which each node has at most one incident
509edge.
[209]510
[40]511There are several different algorithms for calculate matchings in
512graphs.  The matching problems in bipartite graphs are generally
513easier than in general graphs. The goal of the matching optimization
[422]514can be finding maximum cardinality, maximum weight or minimum cost
[40]515matching. The search can be constrained to find perfect or
516maximum cardinality matching.
517
[422]518The 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.
[40]536
537\image html bipartite_matching.png
538\image latex bipartite_matching.eps "Bipartite Matching" width=\textwidth
539*/
540
541/**
[314]542@defgroup spantree Minimum Spanning Tree Algorithms
[40]543@ingroup algs
[50]544\brief Algorithms for finding a minimum cost spanning tree in a graph.
[40]545
[606]546This group contains the algorithms for finding a minimum cost spanning
[422]547tree in a graph.
[40]548*/
549
550/**
[314]551@defgroup auxalg Auxiliary Algorithms
[40]552@ingroup algs
[50]553\brief Auxiliary algorithms implemented in LEMON.
[40]554
[606]555This group contains some algorithms implemented in LEMON
[50]556in order to make it easier to implement complex algorithms.
[40]557*/
558
559/**
[314]560@defgroup approx Approximation Algorithms
561@ingroup algs
[50]562\brief Approximation algorithms.
[40]563
[606]564This group contains the approximation and heuristic algorithms
[50]565implemented in LEMON.
[40]566*/
567
568/**
569@defgroup gen_opt_group General Optimization Tools
[606]570\brief This group contains some general optimization frameworks
[40]571implemented in LEMON.
572
[606]573This group contains some general optimization frameworks
[40]574implemented in LEMON.
575*/
576
577/**
[314]578@defgroup lp_group Lp and Mip Solvers
[40]579@ingroup gen_opt_group
580\brief Lp and Mip solver interfaces for LEMON.
581
[606]582This group contains Lp and Mip solver interfaces for LEMON. The
[40]583various LP solvers could be used in the same manner with this
584interface.
585*/
586
[209]587/**
[314]588@defgroup lp_utils Tools for Lp and Mip Solvers
[40]589@ingroup lp_group
[50]590\brief Helper tools to the Lp and Mip solvers.
[40]591
592This group adds some helper tools to general optimization framework
593implemented in LEMON.
594*/
595
596/**
597@defgroup metah Metaheuristics
598@ingroup gen_opt_group
599\brief Metaheuristics for LEMON library.
600
[606]601This group contains some metaheuristic optimization tools.
[40]602*/
603
604/**
[209]605@defgroup utils Tools and Utilities
[50]606\brief Tools and utilities for programming in LEMON
[40]607
[50]608Tools and utilities for programming in LEMON.
[40]609*/
610
611/**
612@defgroup gutils Basic Graph Utilities
613@ingroup utils
[50]614\brief Simple basic graph utilities.
[40]615
[606]616This group contains some simple basic graph utilities.
[40]617*/
618
619/**
620@defgroup misc Miscellaneous Tools
621@ingroup utils
[50]622\brief Tools for development, debugging and testing.
623
[606]624This group contains several useful tools for development,
[40]625debugging and testing.
626*/
627
628/**
[314]629@defgroup timecount Time Measuring and Counting
[40]630@ingroup misc
[50]631\brief Simple tools for measuring the performance of algorithms.
632
[606]633This group contains simple tools for measuring the performance
[40]634of algorithms.
635*/
636
637/**
638@defgroup exceptions Exceptions
639@ingroup utils
[50]640\brief Exceptions defined in LEMON.
641
[606]642This group contains the exceptions defined in LEMON.
[40]643*/
644
645/**
646@defgroup io_group Input-Output
[50]647\brief Graph Input-Output methods
[40]648
[606]649This group contains the tools for importing and exporting graphs
[314]650and graph related data. Now it supports the \ref lgf-format
651"LEMON Graph Format", the \c DIMACS format and the encapsulated
652postscript (EPS) format.
[40]653*/
654
655/**
[363]656@defgroup lemon_io LEMON Graph Format
[40]657@ingroup io_group
[314]658\brief Reading and writing LEMON Graph Format.
[40]659
[606]660This group contains methods for reading and writing
[236]661\ref lgf-format "LEMON Graph Format".
[40]662*/
663
664/**
[314]665@defgroup eps_io Postscript Exporting
[40]666@ingroup io_group
667\brief General \c EPS drawer and graph exporter
668
[606]669This group contains general \c EPS drawing methods and special
[209]670graph exporting tools.
[40]671*/
672
673/**
[403]674@defgroup dimacs_group DIMACS format
675@ingroup io_group
676\brief Read and write files in DIMACS format
677
678Tools to read a digraph from or write it to a file in DIMACS format data.
679*/
680
681/**
[363]682@defgroup nauty_group NAUTY Format
683@ingroup io_group
684\brief Read \e Nauty format
[403]685
[363]686Tool to read graphs from \e Nauty format data.
687*/
688
689/**
[40]690@defgroup concept Concepts
691\brief Skeleton classes and concept checking classes
692
[606]693This group contains the data/algorithm skeletons and concept checking
[40]694classes implemented in LEMON.
695
696The purpose of the classes in this group is fourfold.
[209]697
[318]698- These classes contain the documentations of the %concepts. In order
[40]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
[318]703  implementation of the %concepts should provide, however completely
[40]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>
[50]712  that makes it possible to check whether a certain implementation of a
[40]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
[606]723This group contains the skeletons and concept checking classes of LEMON's
[40]724graph structures and helper classes used to implement these.
725*/
726
[314]727/**
728@defgroup map_concepts Map Concepts
729@ingroup concept
730\brief Skeleton and concept checking classes for maps
731
[606]732This group contains the skeletons and concept checking classes of maps.
[40]733*/
734
735/**
736\anchor demoprograms
737
[422]738@defgroup demos Demo Programs
[40]739
740Some demo programs are listed here. Their full source codes can be found in
741the \c demo subdirectory of the source tree.
742
[611]743In order to compile them, use the <tt>make demo</tt> or the
744<tt>make check</tt> commands.
[40]745*/
746
747/**
[422]748@defgroup tools Standalone Utility Applications
[40]749
[209]750Some utility applications are listed here.
[40]751
752The standard compilation procedure (<tt>./configure;make</tt>) will compile
[209]753them, as well.
[40]754*/
755
[422]756}
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