COIN-OR::LEMON - Graph Library

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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
19namespace lemon {
20
21/**
22@defgroup datas Data Structures
23This 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
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.
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
43some graph features like arc/edge or node deletion.
44
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
49arcs have to be hidden or the reverse oriented graph have to be used, then
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.
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
59with 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
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
115Since the adaptor classes conform to the \ref graph_concepts "graph concepts",
116an adaptor can even be applied to another one.
117The following image illustrates a situation when a \ref SubDigraph adaptor
118is 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
123The behavior of graph adaptors can be very different. Some of them keep
124capabilities of the original graph while in other cases this would be
125meaningless. This means that the concepts that they meet depend
126on the graph adaptor, and the wrapped graph.
127For example, if an arc of a reversed digraph is deleted, this is carried
128out by deleting the corresponding arc of the original digraph, thus the
129adaptor modifies the original digraph.
130However in case of a residual digraph, this operation has no sense.
131
132Let us stand one more example here to simplify your work.
133ReverseDigraph has constructor
134\code
135ReverseDigraph(Digraph& digraph);
136\endcode
137This means that in a situation, when a <tt>const %ListDigraph&</tt>
138reference to a graph is given, then it have to be instantiated with
139<tt>Digraph=const %ListDigraph</tt>.
140\code
141int 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
153This group contains the map structures implemented in LEMON.
154
155LEMON provides several special purpose maps and map adaptors that e.g. combine
156new 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
166This group contains maps that are specifically designed to assign
167values to the nodes and arcs/edges of graphs.
168
169If 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
178This group contains map adaptors that are used to create "implicit"
179maps from other maps.
180
181Most of them are \ref concepts::ReadMap "read-only maps".
182They 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
186The typical usage of this classes is passing implicit maps to
187algorithms.  If a function type algorithm is called then the function
188type map adaptors can be used comfortable. For example let's see the
189usage 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
208The \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
210and the previously created map. The composed map is a proper function to
211get the color of each node.
212
213The usage with class type algorithms is little bit harder. In this
214case the function type map adaptors can not be used, because the
215function 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
229We have a length map and a maximum speed map on the arcs of a digraph.
230The minimum time to pass the arc can be calculated as the division of
231the two maps which can be done implicitly with the \c DivMap template
232class. 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
241This group contains the path structures implemented in LEMON.
242
243LEMON provides flexible data structures to work with paths.
244All of them have similar interfaces and they can be copied easily with
245assignment operators and copy constructors. This makes it easy and
246efficient to have e.g. the Dijkstra algorithm to store its result in
247any 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
257This group contains the heap structures implemented in LEMON.
258
259LEMON provides several heap classes. They are efficient implementations
260of the abstract data type \e priority \e queue. They store items with
261specified values called \e priorities in such a way that finding and
262removing the item with minimum priority are efficient.
263The basic operations are adding and erasing items, changing the priority
264of an item, etc.
265
266Heaps are crucial in several algorithms, such as Dijkstra and Prim.
267The heap implementations have the same interface, thus any of them can be
268used 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
278This group contains some data structures implemented in LEMON in
279order 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
287This 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
301This group contains two dimensional data storages implemented in LEMON.
302*/
303
304/**
305@defgroup algs Algorithms
306\brief This group contains the several algorithms
307implemented in LEMON.
308
309This group contains the several algorithms
310implemented in LEMON.
311*/
312
313/**
314@defgroup search Graph Search
315@ingroup algs
316\brief Common graph search algorithms.
317
318This 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
328This 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
350This group contains the algorithms for finding minimum cost spanning
351trees 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
359This group contains the algorithms for finding maximum flows and
360feasible circulations \cite clrs01algorithms, \cite amo93networkflows.
361
362The \e maximum \e flow \e problem is to find a flow of maximum value between
363a single source and a single target. Formally, there is a \f$G=(V,A)\f$
364digraph, a \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function and
365\f$s, t \in V\f$ source and target nodes.
366A maximum flow is an \f$f: A\rightarrow\mathbf{R}^+_0\f$ solution of the
367following 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
374LEMON 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
384In most cases the \ref Preflow algorithm provides the
385fastest method for computing a maximum flow. All implementations
386also provide functions to query the minimum cut, which is the dual
387problem of maximum flow.
388
389\ref Circulation is a preflow push-relabel algorithm implemented directly
390for finding feasible circulations, which is a somewhat different problem,
391but it is strongly related to maximum flow.
392For 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
401This group contains the algorithms for finding minimum cost flows and
402circulations \cite amo93networkflows. For more information about this
403problem and its dual solution, see: \ref min_cost_flow
404"Minimum Cost Flow Problem".
405
406LEMON 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
417In general, \ref NetworkSimplex and \ref CostScaling are the most efficient
418implementations.
419\ref NetworkSimplex is usually the fastest on relatively small graphs (up to
420several thousands of nodes) and on dense graphs, while \ref CostScaling is
421typically more efficient on large graphs (e.g. hundreds of thousands of
422nodes or above), especially if they are sparse.
423However, other algorithms could be faster in special cases.
424For example, if the total supply and/or capacities are rather small,
425\ref CapacityScaling is usually the fastest algorithm
426(without effective scaling).
427
428These classes are intended to be used with integer-valued input data
429(capacities, supply values, and costs), except for \ref CapacityScaling,
430which is capable of handling real-valued arc costs (other numerical
431data are required to be integer).
432
433For more details about these implementations and for a comprehensive
434experimental study, see the paper \cite KiralyKovacs12MCF.
435It also compares these codes to other publicly available
436minimum 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
445This group contains the algorithms for finding minimum cut in graphs.
446
447The \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
449outgoing 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
451cut 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
456LEMON 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
465If you want to find minimum cut just between two distinict nodes,
466see 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
474This group contains the algorithms for finding minimum mean cycles
475\cite amo93networkflows, \cite karp78characterization.
476
477The \e minimum \e mean \e cycle \e problem is to find a directed cycle
478of minimum mean length (cost) in a digraph.
479The mean length of a cycle is the average length of its arcs, i.e. the
480ratio between the total length of the cycle and the number of arcs on it.
481
482This 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
484conservative if and only if there is no directed cycle of negative total
485length. For an arbitrary length function, the negative of the minimum
486cycle mean is the smallest \f$\epsilon\f$ value so that increasing the
487arc lengths uniformly by \f$\epsilon\f$ results in a conservative length
488function.
489
490LEMON 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
497In practice, the \ref HowardMmc "Howard" algorithm turned out to be by far the
498most efficient one, though the best known theoretical bound on its running
499time is exponential.
500Both \ref KarpMmc "Karp" and \ref HartmannOrlinMmc "Hartmann-Orlin" algorithms
501run 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
509This group contains the algorithms for calculating
510matchings in graphs and bipartite graphs. The general matching problem is
511finding a subset of the edges for which each node has at most one incident
512edge.
513
514There are several different algorithms for calculate matchings in
515graphs.  The matching problems in bipartite graphs are generally
516easier than in general graphs. The goal of the matching optimization
517can be finding maximum cardinality, maximum weight or minimum cost
518matching. The search can be constrained to find perfect or
519maximum cardinality matching.
520
521The 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
556This group contains the algorithms for discovering the graph properties
557like 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
568This group contains algorithms for finding isomorph copies of a
569given graph in another one, or simply check whether two graphs are isomorphic.
570
571The formal definition of subgraph isomorphism is as follows.
572
573We are given two graphs, \f$G_1=(V_1,E_1)\f$ and \f$G_2=(V_2,E_2)\f$. A
574function \f$f:V_1\longrightarrow V_2\f$ is called \e mapping or \e
575embedding if \f$f(u)\neq f(v)\f$ whenever \f$u\neq v\f$.
576
577The standard <em>Subgraph Isomorphism Problem (SIP)</em> looks for a
578mapping with the property that whenever \f$(u,v)\in E_1\f$, then
579\f$(f(u),f(v))\in E_2\f$.
580
581In case of <em>Induced Subgraph Isomorphism Problem (ISIP)</em> one
582also requires that if \f$(u,v)\not\in E_1\f$, then \f$(f(u),f(v))\not\in
583E_2\f$
584
585In addition, the graph nodes may be \e labeled, i.e. we are given two
586node labelings \f$l_1:V_1\longrightarrow L\f$ and \f$l_2:V_2\longrightarrow
587L\f$ and we require that \f$l_1(u)=l_2(f(u))\f$ holds for all nodes \f$u \in
588G_1\f$.
589
590*/
591
592/**
593@defgroup planar Planar Embedding and Drawing
594@ingroup algs
595\brief Algorithms for planarity checking, embedding and drawing
596
597This group contains the algorithms for planarity checking,
598embedding 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
609This group contains basic heuristic algorithms for the the symmetric
610\e traveling \e salesman \e problem (TSP).
611Given an \ref FullGraph "undirected full graph" with a cost map on its edges,
612the problem is to find a shortest possible tour that visits each node exactly
613once (i.e. the minimum cost Hamiltonian cycle).
614
615These 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.
617Otherwise the algorithms could yield worse results.
618
619LEMON 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
627solution methods. Furthermore, \ref InsertionTsp is usually quite effective.
628
629\ref ChristofidesTsp is somewhat slower, but it has the best guaranteed
630approximation factor: 3/2.
631
632\ref Opt2Tsp usually provides the best results in practice, but
633it is the slowest method. It can also be used to improve given tours,
634for 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
645This group contains the approximation and heuristic algorithms
646implemented 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
658This group contains some algorithms implemented in LEMON
659in 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
665implemented in LEMON.
666
667This group contains some general optimization frameworks
668implemented 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
676This group contains LP and MIP solver interfaces for LEMON.
677Various LP solvers could be used in the same manner with this
678high-level interface.
679
680The 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
689This group adds some helper tools to general optimization framework
690implemented in LEMON.
691*/
692
693/**
694@defgroup metah Metaheuristics
695@ingroup gen_opt_group
696\brief Metaheuristics for LEMON library.
697
698This 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
705Tools 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
713This 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
721This group contains several useful tools for development,
722debugging 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
730This group contains simple tools for measuring the performance
731of algorithms.
732*/
733
734/**
735@defgroup exceptions Exceptions
736@ingroup utils
737\brief Exceptions defined in LEMON.
738
739This group contains the exceptions defined in LEMON.
740*/
741
742/**
743@defgroup io_group Input-Output
744\brief Graph Input-Output methods
745
746This group contains the tools for importing and exporting graphs
747and graph related data. Now it supports the \ref lgf-format
748"LEMON Graph Format", the \c DIMACS format and the encapsulated
749postscript (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
757This 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
766This group contains general \c EPS drawing methods and special
767graph 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
777Tools 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
785Tool to read graphs from \e Nauty format data.
786*/
787
788/**
789@defgroup concept Concepts
790\brief Skeleton classes and concept checking classes
791
792This group contains the data/algorithm skeletons and concept checking
793classes implemented in LEMON.
794
795The 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
822This group contains the skeletons and concept checking classes of
823graph structures.
824*/
825
826/**
827@defgroup map_concepts Map Concepts
828@ingroup concept
829\brief Skeleton and concept checking classes for maps
830
831This group contains the skeletons and concept checking classes of maps.
832*/
833
834/**
835@defgroup tools Standalone Utility Applications
836
837Some utility applications are listed here.
838
839The standard compilation procedure (<tt>./configure;make</tt>) will compile
840them, as well.
841*/
842
843/**
844\anchor demoprograms
845
846@defgroup demos Demo Programs
847
848Some demo programs are listed here. Their full source codes can be found in
849the \c demo subdirectory of the source tree.
850
851In order to compile them, use the <tt>make demo</tt> or the
852<tt>make check</tt> commands.
853*/
854
855}
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