COIN-OR::LEMON - Graph Library

source: lemon/doc/groups.dox @ 843:189760a7cdd0

1.1
Last change on this file since 843:189760a7cdd0 was 843:189760a7cdd0, checked in by Peter Kovacs <kpeter@…>, 11 years ago

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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) 2003-2009
6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
7 * (Egervary Research Group on Combinatorial Optimization, EGRES).
8 *
9 * Permission to use, modify and distribute this software is granted
10 * provided that this copyright notice appears in all copies. For
11 * precise terms see the accompanying LICENSE file.
12 *
13 * This software is provided "AS IS" with no warranty of any kind,
14 * express or implied, and with no claim as to its suitability for any
15 * purpose.
16 *
17 */
18
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
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
138*/
139
140/**
141@defgroup semi_adaptors Semi-Adaptor Classes for Graphs
142@ingroup graphs
143\brief Graph types between real graphs and graph adaptors.
144
145This group contains some graph types between real graphs and graph adaptors.
146These classes wrap graphs to give new functionality as the adaptors do it.
147On the other hand they are not light-weight structures as the adaptors.
148*/
149
150/**
151@defgroup maps Maps
152@ingroup datas
153\brief Map structures implemented in LEMON.
154
155This group contains the map structures implemented in LEMON.
156
157LEMON provides several special purpose maps and map adaptors that e.g. combine
158new maps from existing ones.
159
160<b>See also:</b> \ref map_concepts "Map Concepts".
161*/
162
163/**
164@defgroup graph_maps Graph Maps
165@ingroup maps
166\brief Special graph-related maps.
167
168This group contains maps that are specifically designed to assign
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".
173*/
174
175/**
176\defgroup map_adaptors Map Adaptors
177\ingroup maps
178\brief Tools to create new maps from existing ones
179
180This group contains map adaptors that are used to create "implicit"
181maps from other maps.
182
183Most of them are \ref concepts::ReadMap "read-only maps".
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.
187
188The typical usage of this classes is passing implicit maps to
189algorithms.  If a function type algorithm is called then the function
190type map adaptors can be used comfortable. For example let's see the
191usage of map adaptors with the \c graphToEps() function.
192\code
193  Color nodeColor(int deg) {
194    if (deg >= 2) {
195      return Color(0.5, 0.0, 0.5);
196    } else if (deg == 1) {
197      return Color(1.0, 0.5, 1.0);
198    } else {
199      return Color(0.0, 0.0, 0.0);
200    }
201  }
202
203  Digraph::NodeMap<int> degree_map(graph);
204
205  graphToEps(graph, "graph.eps")
206    .coords(coords).scaleToA4().undirected()
207    .nodeColors(composeMap(functorToMap(nodeColor), degree_map))
208    .run();
209\endcode
210The \c functorToMap() function makes an \c int to \c Color map from the
211\c nodeColor() function. The \c composeMap() compose the \c degree_map
212and the previously created map. The composed map is a proper function to
213get the color of each node.
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
217function map adaptors give back temporary objects.
218\code
219  Digraph graph;
220
221  typedef Digraph::ArcMap<double> DoubleArcMap;
222  DoubleArcMap length(graph);
223  DoubleArcMap speed(graph);
224
225  typedef DivMap<DoubleArcMap, DoubleArcMap> TimeMap;
226  TimeMap time(length, speed);
227
228  Dijkstra<Digraph, TimeMap> dijkstra(graph, time);
229  dijkstra.run(source, target);
230\endcode
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
234class. We use the implicit minimum time map as the length map of the
235\c Dijkstra algorithm.
236*/
237
238/**
239@defgroup paths Path Structures
240@ingroup datas
241\brief %Path structures implemented in LEMON.
242
243This group contains the path structures implemented in LEMON.
244
245LEMON provides flexible data structures to work with paths.
246All of them have similar interfaces and they can be copied easily with
247assignment operators and copy constructors. This makes it easy and
248efficient to have e.g. the Dijkstra algorithm to store its result in
249any kind of path structure.
250
251\sa lemon::concepts::Path
252*/
253
254/**
255@defgroup auxdat Auxiliary Data Structures
256@ingroup datas
257\brief Auxiliary data structures implemented in LEMON.
258
259This group contains some data structures implemented in LEMON in
260order to make it easier to implement combinatorial algorithms.
261*/
262
263/**
264@defgroup algs Algorithms
265\brief This group contains the several algorithms
266implemented in LEMON.
267
268This group contains the several algorithms
269implemented in LEMON.
270*/
271
272/**
273@defgroup search Graph Search
274@ingroup algs
275\brief Common graph search algorithms.
276
277This group contains the common graph search algorithms, namely
278\e breadth-first \e search (BFS) and \e depth-first \e search (DFS).
279*/
280
281/**
282@defgroup shortest_path Shortest Path Algorithms
283@ingroup algs
284\brief Algorithms for finding shortest paths.
285
286This group contains the algorithms for finding shortest paths in digraphs.
287
288 - \ref Dijkstra Dijkstra's algorithm for finding shortest paths from a
289   source node when all arc lengths are non-negative.
290 - \ref Suurballe A successive shortest path algorithm for finding
291   arc-disjoint paths between two nodes having minimum total length.
292*/
293
294/**
295@defgroup max_flow Maximum Flow Algorithms
296@ingroup algs
297\brief Algorithms for finding maximum flows.
298
299This group contains the algorithms for finding maximum flows and
300feasible circulations.
301
302The \e maximum \e flow \e problem is to find a flow of maximum value between
303a single source and a single target. Formally, there is a \f$G=(V,A)\f$
304digraph, a \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function and
305\f$s, t \in V\f$ source and target nodes.
306A maximum flow is an \f$f: A\rightarrow\mathbf{R}^+_0\f$ solution of the
307following optimization problem.
308
309\f[ \max\sum_{sv\in A} f(sv) - \sum_{vs\in A} f(vs) \f]
310\f[ \sum_{uv\in A} f(uv) = \sum_{vu\in A} f(vu)
311    \quad \forall u\in V\setminus\{s,t\} \f]
312\f[ 0 \leq f(uv) \leq cap(uv) \quad \forall uv\in A \f]
313
314\ref Preflow implements the preflow push-relabel algorithm of Goldberg and
315Tarjan for solving this problem. It also provides functions to query the
316minimum cut, which is the dual problem of maximum flow.
317
318\ref Circulation is a preflow push-relabel algorithm implemented directly
319for finding feasible circulations, which is a somewhat different problem,
320but it is strongly related to maximum flow.
321For more information, see \ref Circulation.
322*/
323
324/**
325@defgroup min_cost_flow Minimum Cost Flow Algorithms
326@ingroup algs
327
328\brief Algorithms for finding minimum cost flows and circulations.
329
330This group contains the algorithms for finding minimum cost flows and
331circulations.
332
333The \e minimum \e cost \e flow \e problem is to find a feasible flow of
334minimum total cost from a set of supply nodes to a set of demand nodes
335in a network with capacity constraints (lower and upper bounds)
336and arc costs.
337Formally, let \f$G=(V,A)\f$ be a digraph, \f$lower: A\rightarrow\mathbf{Z}\f$,
338\f$upper: A\rightarrow\mathbf{Z}\cup\{+\infty\}\f$ denote the lower and
339upper bounds for the flow values on the arcs, for which
340\f$lower(uv) \leq upper(uv)\f$ must hold for all \f$uv\in A\f$,
341\f$cost: A\rightarrow\mathbf{Z}\f$ denotes the cost per unit flow
342on the arcs and \f$sup: V\rightarrow\mathbf{Z}\f$ denotes the
343signed supply values of the nodes.
344If \f$sup(u)>0\f$, then \f$u\f$ is a supply node with \f$sup(u)\f$
345supply, if \f$sup(u)<0\f$, then \f$u\f$ is a demand node with
346\f$-sup(u)\f$ demand.
347A minimum cost flow is an \f$f: A\rightarrow\mathbf{Z}\f$ solution
348of the following optimization problem.
349
350\f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f]
351\f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \geq
352    sup(u) \quad \forall u\in V \f]
353\f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f]
354
355The sum of the supply values, i.e. \f$\sum_{u\in V} sup(u)\f$ must be
356zero or negative in order to have a feasible solution (since the sum
357of the expressions on the left-hand side of the inequalities is zero).
358It means that the total demand must be greater or equal to the total
359supply and all the supplies have to be carried out from the supply nodes,
360but there could be demands that are not satisfied.
361If \f$\sum_{u\in V} sup(u)\f$ is zero, then all the supply/demand
362constraints have to be satisfied with equality, i.e. all demands
363have to be satisfied and all supplies have to be used.
364
365If you need the opposite inequalities in the supply/demand constraints
366(i.e. the total demand is less than the total supply and all the demands
367have to be satisfied while there could be supplies that are not used),
368then you could easily transform the problem to the above form by reversing
369the direction of the arcs and taking the negative of the supply values
370(e.g. using \ref ReverseDigraph and \ref NegMap adaptors).
371However \ref NetworkSimplex algorithm also supports this form directly
372for the sake of convenience.
373
374A feasible solution for this problem can be found using \ref Circulation.
375
376Note that the above formulation is actually more general than the usual
377definition of the minimum cost flow problem, in which strict equalities
378are required in the supply/demand contraints, i.e.
379
380\f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) =
381    sup(u) \quad \forall u\in V. \f]
382
383However if the sum of the supply values is zero, then these two problems
384are equivalent. So if you need the equality form, you have to ensure this
385additional contraint for the algorithms.
386
387The dual solution of the minimum cost flow problem is represented by node
388potentials \f$\pi: V\rightarrow\mathbf{Z}\f$.
389An \f$f: A\rightarrow\mathbf{Z}\f$ feasible solution of the problem
390is optimal if and only if for some \f$\pi: V\rightarrow\mathbf{Z}\f$
391node potentials the following \e complementary \e slackness optimality
392conditions hold.
393
394 - For all \f$uv\in A\f$ arcs:
395   - if \f$cost^\pi(uv)>0\f$, then \f$f(uv)=lower(uv)\f$;
396   - if \f$lower(uv)<f(uv)<upper(uv)\f$, then \f$cost^\pi(uv)=0\f$;
397   - if \f$cost^\pi(uv)<0\f$, then \f$f(uv)=upper(uv)\f$.
398 - For all \f$u\in V\f$ nodes:
399   - if \f$\sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \neq sup(u)\f$,
400     then \f$\pi(u)=0\f$.
401 
402Here \f$cost^\pi(uv)\f$ denotes the \e reduced \e cost of the arc
403\f$uv\in A\f$ with respect to the potential function \f$\pi\f$, i.e.
404\f[ cost^\pi(uv) = cost(uv) + \pi(u) - \pi(v).\f]
405
406\ref NetworkSimplex is an efficient implementation of the primal Network
407Simplex algorithm for finding minimum cost flows. It also provides dual
408solution (node potentials), if an optimal flow is found.
409*/
410
411/**
412@defgroup min_cut Minimum Cut Algorithms
413@ingroup algs
414
415\brief Algorithms for finding minimum cut in graphs.
416
417This group contains the algorithms for finding minimum cut in graphs.
418
419The \e minimum \e cut \e problem is to find a non-empty and non-complete
420\f$X\f$ subset of the nodes with minimum overall capacity on
421outgoing arcs. Formally, there is a \f$G=(V,A)\f$ digraph, a
422\f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function. The minimum
423cut is the \f$X\f$ solution of the next optimization problem:
424
425\f[ \min_{X \subset V, X\not\in \{\emptyset, V\}}
426    \sum_{uv\in A, u\in X, v\not\in X}cap(uv) \f]
427
428LEMON contains several algorithms related to minimum cut problems:
429
430- \ref HaoOrlin "Hao-Orlin algorithm" for calculating minimum cut
431  in directed graphs.
432- \ref GomoryHu "Gomory-Hu tree computation" for calculating
433  all-pairs minimum cut in undirected graphs.
434
435If you want to find minimum cut just between two distinict nodes,
436see the \ref max_flow "maximum flow problem".
437*/
438
439/**
440@defgroup graph_properties Connectivity and Other Graph Properties
441@ingroup algs
442\brief Algorithms for discovering the graph properties
443
444This group contains the algorithms for discovering the graph properties
445like connectivity, bipartiteness, euler property, simplicity etc.
446
447\image html edge_biconnected_components.png
448\image latex edge_biconnected_components.eps "bi-edge-connected components" width=\textwidth
449*/
450
451/**
452@defgroup matching Matching Algorithms
453@ingroup algs
454\brief Algorithms for finding matchings in graphs and bipartite graphs.
455
456This group contains the algorithms for calculating matchings in graphs.
457The general matching problem is finding a subset of the edges for which
458each node has at most one incident edge.
459
460There are several different algorithms for calculate matchings in
461graphs. The goal of the matching optimization
462can be finding maximum cardinality, maximum weight or minimum cost
463matching. The search can be constrained to find perfect or
464maximum cardinality matching.
465
466The matching algorithms implemented in LEMON:
467- \ref MaxMatching Edmond's blossom shrinking algorithm for calculating
468  maximum cardinality matching in general graphs.
469- \ref MaxWeightedMatching Edmond's blossom shrinking algorithm for calculating
470  maximum weighted matching in general graphs.
471- \ref MaxWeightedPerfectMatching
472  Edmond's blossom shrinking algorithm for calculating maximum weighted
473  perfect matching in general graphs.
474
475\image html bipartite_matching.png
476\image latex bipartite_matching.eps "Bipartite Matching" width=\textwidth
477*/
478
479/**
480@defgroup spantree Minimum Spanning Tree Algorithms
481@ingroup algs
482\brief Algorithms for finding a minimum cost spanning tree in a graph.
483
484This group contains the algorithms for finding a minimum cost spanning
485tree in a graph.
486*/
487
488/**
489@defgroup auxalg Auxiliary Algorithms
490@ingroup algs
491\brief Auxiliary algorithms implemented in LEMON.
492
493This group contains some algorithms implemented in LEMON
494in order to make it easier to implement complex algorithms.
495*/
496
497/**
498@defgroup gen_opt_group General Optimization Tools
499\brief This group contains some general optimization frameworks
500implemented in LEMON.
501
502This group contains some general optimization frameworks
503implemented in LEMON.
504*/
505
506/**
507@defgroup lp_group Lp and Mip Solvers
508@ingroup gen_opt_group
509\brief Lp and Mip solver interfaces for LEMON.
510
511This group contains Lp and Mip solver interfaces for LEMON. The
512various LP solvers could be used in the same manner with this
513interface.
514*/
515
516/**
517@defgroup utils Tools and Utilities
518\brief Tools and utilities for programming in LEMON
519
520Tools and utilities for programming in LEMON.
521*/
522
523/**
524@defgroup gutils Basic Graph Utilities
525@ingroup utils
526\brief Simple basic graph utilities.
527
528This group contains some simple basic graph utilities.
529*/
530
531/**
532@defgroup misc Miscellaneous Tools
533@ingroup utils
534\brief Tools for development, debugging and testing.
535
536This group contains several useful tools for development,
537debugging and testing.
538*/
539
540/**
541@defgroup timecount Time Measuring and Counting
542@ingroup misc
543\brief Simple tools for measuring the performance of algorithms.
544
545This group contains simple tools for measuring the performance
546of algorithms.
547*/
548
549/**
550@defgroup exceptions Exceptions
551@ingroup utils
552\brief Exceptions defined in LEMON.
553
554This group contains the exceptions defined in LEMON.
555*/
556
557/**
558@defgroup io_group Input-Output
559\brief Graph Input-Output methods
560
561This group contains the tools for importing and exporting graphs
562and graph related data. Now it supports the \ref lgf-format
563"LEMON Graph Format", the \c DIMACS format and the encapsulated
564postscript (EPS) format.
565*/
566
567/**
568@defgroup lemon_io LEMON Graph Format
569@ingroup io_group
570\brief Reading and writing LEMON Graph Format.
571
572This group contains methods for reading and writing
573\ref lgf-format "LEMON Graph Format".
574*/
575
576/**
577@defgroup eps_io Postscript Exporting
578@ingroup io_group
579\brief General \c EPS drawer and graph exporter
580
581This group contains general \c EPS drawing methods and special
582graph exporting tools.
583*/
584
585/**
586@defgroup dimacs_group DIMACS format
587@ingroup io_group
588\brief Read and write files in DIMACS format
589
590Tools to read a digraph from or write it to a file in DIMACS format data.
591*/
592
593/**
594@defgroup nauty_group NAUTY Format
595@ingroup io_group
596\brief Read \e Nauty format
597
598Tool to read graphs from \e Nauty format data.
599*/
600
601/**
602@defgroup concept Concepts
603\brief Skeleton classes and concept checking classes
604
605This group contains the data/algorithm skeletons and concept checking
606classes implemented in LEMON.
607
608The purpose of the classes in this group is fourfold.
609
610- These classes contain the documentations of the %concepts. In order
611  to avoid document multiplications, an implementation of a concept
612  simply refers to the corresponding concept class.
613
614- These classes declare every functions, <tt>typedef</tt>s etc. an
615  implementation of the %concepts should provide, however completely
616  without implementations and real data structures behind the
617  interface. On the other hand they should provide nothing else. All
618  the algorithms working on a data structure meeting a certain concept
619  should compile with these classes. (Though it will not run properly,
620  of course.) In this way it is easily to check if an algorithm
621  doesn't use any extra feature of a certain implementation.
622
623- The concept descriptor classes also provide a <em>checker class</em>
624  that makes it possible to check whether a certain implementation of a
625  concept indeed provides all the required features.
626
627- Finally, They can serve as a skeleton of a new implementation of a concept.
628*/
629
630/**
631@defgroup graph_concepts Graph Structure Concepts
632@ingroup concept
633\brief Skeleton and concept checking classes for graph structures
634
635This group contains the skeletons and concept checking classes of LEMON's
636graph structures and helper classes used to implement these.
637*/
638
639/**
640@defgroup map_concepts Map Concepts
641@ingroup concept
642\brief Skeleton and concept checking classes for maps
643
644This group contains the skeletons and concept checking classes of maps.
645*/
646
647/**
648\anchor demoprograms
649
650@defgroup demos Demo Programs
651
652Some demo programs are listed here. Their full source codes can be found in
653the \c demo subdirectory of the source tree.
654
655In order to compile them, use the <tt>make demo</tt> or the
656<tt>make check</tt> commands.
657*/
658
659/**
660@defgroup tools Standalone Utility Applications
661
662Some utility applications are listed here.
663
664The standard compilation procedure (<tt>./configure;make</tt>) will compile
665them, as well.
666*/
667
668}
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