COIN-OR::LEMON - Graph Library

source: lemon/doc/groups.dox @ 710:8b0df68370a4

Last change on this file since 710:8b0df68370a4 was 710:8b0df68370a4, checked in by Peter Kovacs <kpeter@…>, 10 years ago

Fix the GEQ/LEQ handling in NetworkSimplex? + improve doc (#291)

  • Fix the optimality conditions for the GEQ/LEQ form.
  • Fix the initialization of the algortihm. It ensures correct solutions and it is much faster for the inequality forms.
  • Fix the pivot rules to search all the arcs that have to be allowed to get in the basis.
  • Better block size for the Block Search pivot rule.
  • Improve documentation of the problem and move it to a separate page.
File size: 22.3 KB
Line 
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 maps Maps
142@ingroup datas
143\brief Map structures implemented in LEMON.
144
145This group contains the map structures implemented in LEMON.
146
147LEMON provides several special purpose maps and map adaptors that e.g. combine
148new maps from existing ones.
149
150<b>See also:</b> \ref map_concepts "Map Concepts".
151*/
152
153/**
154@defgroup graph_maps Graph Maps
155@ingroup maps
156\brief Special graph-related maps.
157
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".
163*/
164
165/**
166\defgroup map_adaptors Map Adaptors
167\ingroup maps
168\brief Tools to create new maps from existing ones
169
170This group contains map adaptors that are used to create "implicit"
171maps from other maps.
172
173Most of them are \ref concepts::ReadMap "read-only maps".
174They can make arithmetic and logical operations between one or two maps
175(negation, shifting, addition, multiplication, logical 'and', 'or',
176'not' etc.) or e.g. convert a map to another one of different Value type.
177
178The typical usage of this classes is passing implicit maps to
179algorithms.  If a function type algorithm is called then the function
180type map adaptors can be used comfortable. For example let's see the
181usage of map adaptors with the \c graphToEps() function.
182\code
183  Color nodeColor(int deg) {
184    if (deg >= 2) {
185      return Color(0.5, 0.0, 0.5);
186    } else if (deg == 1) {
187      return Color(1.0, 0.5, 1.0);
188    } else {
189      return Color(0.0, 0.0, 0.0);
190    }
191  }
192
193  Digraph::NodeMap<int> degree_map(graph);
194
195  graphToEps(graph, "graph.eps")
196    .coords(coords).scaleToA4().undirected()
197    .nodeColors(composeMap(functorToMap(nodeColor), degree_map))
198    .run();
199\endcode
200The \c functorToMap() function makes an \c int to \c Color map from the
201\c nodeColor() function. The \c composeMap() compose the \c degree_map
202and the previously created map. The composed map is a proper function to
203get the color of each node.
204
205The usage with class type algorithms is little bit harder. In this
206case the function type map adaptors can not be used, because the
207function map adaptors give back temporary objects.
208\code
209  Digraph graph;
210
211  typedef Digraph::ArcMap<double> DoubleArcMap;
212  DoubleArcMap length(graph);
213  DoubleArcMap speed(graph);
214
215  typedef DivMap<DoubleArcMap, DoubleArcMap> TimeMap;
216  TimeMap time(length, speed);
217
218  Dijkstra<Digraph, TimeMap> dijkstra(graph, time);
219  dijkstra.run(source, target);
220\endcode
221We have a length map and a maximum speed map on the arcs of a digraph.
222The minimum time to pass the arc can be calculated as the division of
223the two maps which can be done implicitly with the \c DivMap template
224class. We use the implicit minimum time map as the length map of the
225\c Dijkstra algorithm.
226*/
227
228/**
229@defgroup matrices Matrices
230@ingroup datas
231\brief Two dimensional data storages implemented in LEMON.
232
233This group contains two dimensional data storages implemented in LEMON.
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 lemon::concepts::Path
250*/
251
252/**
253@defgroup auxdat Auxiliary Data Structures
254@ingroup datas
255\brief Auxiliary data structures implemented in LEMON.
256
257This group contains some data structures implemented in LEMON in
258order to make it easier to implement combinatorial algorithms.
259*/
260
261/**
262@defgroup algs Algorithms
263\brief This group contains the several algorithms
264implemented in LEMON.
265
266This group contains the several algorithms
267implemented in LEMON.
268*/
269
270/**
271@defgroup search Graph Search
272@ingroup algs
273\brief Common graph search algorithms.
274
275This group contains the common graph search algorithms, namely
276\e breadth-first \e search (BFS) and \e depth-first \e search (DFS).
277*/
278
279/**
280@defgroup shortest_path Shortest Path Algorithms
281@ingroup algs
282\brief Algorithms for finding shortest paths.
283
284This group contains the algorithms for finding shortest paths in digraphs.
285
286 - \ref Dijkstra algorithm for finding shortest paths from a source node
287   when all arc lengths are non-negative.
288 - \ref BellmanFord "Bellman-Ford" algorithm for finding shortest paths
289   from a source node when arc lenghts can be either positive or negative,
290   but the digraph should not contain directed cycles with negative total
291   length.
292 - \ref FloydWarshall "Floyd-Warshall" and \ref Johnson "Johnson" algorithms
293   for solving the \e all-pairs \e shortest \e paths \e problem when arc
294   lenghts can be either positive or negative, but the digraph should
295   not contain directed cycles with negative total length.
296 - \ref Suurballe A successive shortest path algorithm for finding
297   arc-disjoint paths between two nodes having minimum total length.
298*/
299
300/**
301@defgroup max_flow Maximum Flow Algorithms
302@ingroup algs
303\brief Algorithms for finding maximum flows.
304
305This group contains the algorithms for finding maximum flows and
306feasible circulations.
307
308The \e maximum \e flow \e problem is to find a flow of maximum value between
309a single source and a single target. Formally, there is a \f$G=(V,A)\f$
310digraph, a \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function and
311\f$s, t \in V\f$ source and target nodes.
312A maximum flow is an \f$f: A\rightarrow\mathbf{R}^+_0\f$ solution of the
313following optimization problem.
314
315\f[ \max\sum_{sv\in A} f(sv) - \sum_{vs\in A} f(vs) \f]
316\f[ \sum_{uv\in A} f(uv) = \sum_{vu\in A} f(vu)
317    \quad \forall u\in V\setminus\{s,t\} \f]
318\f[ 0 \leq f(uv) \leq cap(uv) \quad \forall uv\in A \f]
319
320LEMON contains several algorithms for solving maximum flow problems:
321- \ref EdmondsKarp Edmonds-Karp algorithm.
322- \ref Preflow Goldberg-Tarjan's preflow push-relabel algorithm.
323- \ref DinitzSleatorTarjan Dinitz's blocking flow algorithm with dynamic trees.
324- \ref GoldbergTarjan Preflow push-relabel algorithm with dynamic trees.
325
326In most cases the \ref Preflow "Preflow" algorithm provides the
327fastest method for computing a maximum flow. All implementations
328also provide functions to query the minimum cut, which is the dual
329problem of maximum flow.
330
331\ref Circulation is a preflow push-relabel algorithm implemented directly
332for finding feasible circulations, which is a somewhat different problem,
333but it is strongly related to maximum flow.
334For more information, see \ref Circulation.
335*/
336
337/**
338@defgroup min_cost_flow_algs Minimum Cost Flow Algorithms
339@ingroup algs
340
341\brief Algorithms for finding minimum cost flows and circulations.
342
343This group contains the algorithms for finding minimum cost flows and
344circulations. For more information about this problem and its dual
345solution see \ref min_cost_flow "Minimum Cost Flow Problem".
346
347LEMON contains several algorithms for this problem.
348 - \ref NetworkSimplex Primal Network Simplex algorithm with various
349   pivot strategies.
350 - \ref CostScaling Push-Relabel and Augment-Relabel algorithms based on
351   cost scaling.
352 - \ref CapacityScaling Successive Shortest %Path algorithm with optional
353   capacity scaling.
354 - \ref CancelAndTighten The Cancel and Tighten algorithm.
355 - \ref CycleCanceling Cycle-Canceling algorithms.
356
357In general NetworkSimplex is the most efficient implementation,
358but in special cases other algorithms could be faster.
359For example, if the total supply and/or capacities are rather small,
360CapacityScaling is usually the fastest algorithm (without effective scaling).
361*/
362
363/**
364@defgroup min_cut Minimum Cut Algorithms
365@ingroup algs
366
367\brief Algorithms for finding minimum cut in graphs.
368
369This group contains the algorithms for finding minimum cut in graphs.
370
371The \e minimum \e cut \e problem is to find a non-empty and non-complete
372\f$X\f$ subset of the nodes with minimum overall capacity on
373outgoing arcs. Formally, there is a \f$G=(V,A)\f$ digraph, a
374\f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function. The minimum
375cut is the \f$X\f$ solution of the next optimization problem:
376
377\f[ \min_{X \subset V, X\not\in \{\emptyset, V\}}
378    \sum_{uv\in A, u\in X, v\not\in X}cap(uv) \f]
379
380LEMON contains several algorithms related to minimum cut problems:
381
382- \ref HaoOrlin "Hao-Orlin algorithm" for calculating minimum cut
383  in directed graphs.
384- \ref NagamochiIbaraki "Nagamochi-Ibaraki algorithm" for
385  calculating minimum cut in undirected graphs.
386- \ref GomoryHu "Gomory-Hu tree computation" for calculating
387  all-pairs minimum cut in undirected graphs.
388
389If you want to find minimum cut just between two distinict nodes,
390see the \ref max_flow "maximum flow problem".
391*/
392
393/**
394@defgroup graph_properties Connectivity and Other Graph Properties
395@ingroup algs
396\brief Algorithms for discovering the graph properties
397
398This group contains the algorithms for discovering the graph properties
399like connectivity, bipartiteness, euler property, simplicity etc.
400
401\image html edge_biconnected_components.png
402\image latex edge_biconnected_components.eps "bi-edge-connected components" width=\textwidth
403*/
404
405/**
406@defgroup planar Planarity Embedding and Drawing
407@ingroup algs
408\brief Algorithms for planarity checking, embedding and drawing
409
410This group contains the algorithms for planarity checking,
411embedding and drawing.
412
413\image html planar.png
414\image latex planar.eps "Plane graph" width=\textwidth
415*/
416
417/**
418@defgroup matching Matching Algorithms
419@ingroup algs
420\brief Algorithms for finding matchings in graphs and bipartite graphs.
421
422This group contains the algorithms for calculating
423matchings in graphs and bipartite graphs. The general matching problem is
424finding a subset of the edges for which each node has at most one incident
425edge.
426
427There are several different algorithms for calculate matchings in
428graphs.  The matching problems in bipartite graphs are generally
429easier than in general graphs. The goal of the matching optimization
430can be finding maximum cardinality, maximum weight or minimum cost
431matching. The search can be constrained to find perfect or
432maximum cardinality matching.
433
434The matching algorithms implemented in LEMON:
435- \ref MaxBipartiteMatching Hopcroft-Karp augmenting path algorithm
436  for calculating maximum cardinality matching in bipartite graphs.
437- \ref PrBipartiteMatching Push-relabel algorithm
438  for calculating maximum cardinality matching in bipartite graphs.
439- \ref MaxWeightedBipartiteMatching
440  Successive shortest path algorithm for calculating maximum weighted
441  matching and maximum weighted bipartite matching in bipartite graphs.
442- \ref MinCostMaxBipartiteMatching
443  Successive shortest path algorithm for calculating minimum cost maximum
444  matching in bipartite graphs.
445- \ref MaxMatching Edmond's blossom shrinking algorithm for calculating
446  maximum cardinality matching in general graphs.
447- \ref MaxWeightedMatching Edmond's blossom shrinking algorithm for calculating
448  maximum weighted matching in general graphs.
449- \ref MaxWeightedPerfectMatching
450  Edmond's blossom shrinking algorithm for calculating maximum weighted
451  perfect matching in general graphs.
452
453\image html bipartite_matching.png
454\image latex bipartite_matching.eps "Bipartite Matching" width=\textwidth
455*/
456
457/**
458@defgroup spantree Minimum Spanning Tree Algorithms
459@ingroup algs
460\brief Algorithms for finding minimum cost spanning trees and arborescences.
461
462This group contains the algorithms for finding minimum cost spanning
463trees and arborescences.
464*/
465
466/**
467@defgroup auxalg Auxiliary Algorithms
468@ingroup algs
469\brief Auxiliary algorithms implemented in LEMON.
470
471This group contains some algorithms implemented in LEMON
472in order to make it easier to implement complex algorithms.
473*/
474
475/**
476@defgroup approx Approximation Algorithms
477@ingroup algs
478\brief Approximation algorithms.
479
480This group contains the approximation and heuristic algorithms
481implemented in LEMON.
482*/
483
484/**
485@defgroup gen_opt_group General Optimization Tools
486\brief This group contains some general optimization frameworks
487implemented in LEMON.
488
489This group contains some general optimization frameworks
490implemented in LEMON.
491*/
492
493/**
494@defgroup lp_group Lp and Mip Solvers
495@ingroup gen_opt_group
496\brief Lp and Mip solver interfaces for LEMON.
497
498This group contains Lp and Mip solver interfaces for LEMON. The
499various LP solvers could be used in the same manner with this
500interface.
501*/
502
503/**
504@defgroup lp_utils Tools for Lp and Mip Solvers
505@ingroup lp_group
506\brief Helper tools to the Lp and Mip solvers.
507
508This group adds some helper tools to general optimization framework
509implemented in LEMON.
510*/
511
512/**
513@defgroup metah Metaheuristics
514@ingroup gen_opt_group
515\brief Metaheuristics for LEMON library.
516
517This group contains some metaheuristic optimization tools.
518*/
519
520/**
521@defgroup utils Tools and Utilities
522\brief Tools and utilities for programming in LEMON
523
524Tools and utilities for programming in LEMON.
525*/
526
527/**
528@defgroup gutils Basic Graph Utilities
529@ingroup utils
530\brief Simple basic graph utilities.
531
532This group contains some simple basic graph utilities.
533*/
534
535/**
536@defgroup misc Miscellaneous Tools
537@ingroup utils
538\brief Tools for development, debugging and testing.
539
540This group contains several useful tools for development,
541debugging and testing.
542*/
543
544/**
545@defgroup timecount Time Measuring and Counting
546@ingroup misc
547\brief Simple tools for measuring the performance of algorithms.
548
549This group contains simple tools for measuring the performance
550of algorithms.
551*/
552
553/**
554@defgroup exceptions Exceptions
555@ingroup utils
556\brief Exceptions defined in LEMON.
557
558This group contains the exceptions defined in LEMON.
559*/
560
561/**
562@defgroup io_group Input-Output
563\brief Graph Input-Output methods
564
565This group contains the tools for importing and exporting graphs
566and graph related data. Now it supports the \ref lgf-format
567"LEMON Graph Format", the \c DIMACS format and the encapsulated
568postscript (EPS) format.
569*/
570
571/**
572@defgroup lemon_io LEMON Graph Format
573@ingroup io_group
574\brief Reading and writing LEMON Graph Format.
575
576This group contains methods for reading and writing
577\ref lgf-format "LEMON Graph Format".
578*/
579
580/**
581@defgroup eps_io Postscript Exporting
582@ingroup io_group
583\brief General \c EPS drawer and graph exporter
584
585This group contains general \c EPS drawing methods and special
586graph exporting tools.
587*/
588
589/**
590@defgroup dimacs_group DIMACS format
591@ingroup io_group
592\brief Read and write files in DIMACS format
593
594Tools to read a digraph from or write it to a file in DIMACS format data.
595*/
596
597/**
598@defgroup nauty_group NAUTY Format
599@ingroup io_group
600\brief Read \e Nauty format
601
602Tool to read graphs from \e Nauty format data.
603*/
604
605/**
606@defgroup concept Concepts
607\brief Skeleton classes and concept checking classes
608
609This group contains the data/algorithm skeletons and concept checking
610classes implemented in LEMON.
611
612The purpose of the classes in this group is fourfold.
613
614- These classes contain the documentations of the %concepts. In order
615  to avoid document multiplications, an implementation of a concept
616  simply refers to the corresponding concept class.
617
618- These classes declare every functions, <tt>typedef</tt>s etc. an
619  implementation of the %concepts should provide, however completely
620  without implementations and real data structures behind the
621  interface. On the other hand they should provide nothing else. All
622  the algorithms working on a data structure meeting a certain concept
623  should compile with these classes. (Though it will not run properly,
624  of course.) In this way it is easily to check if an algorithm
625  doesn't use any extra feature of a certain implementation.
626
627- The concept descriptor classes also provide a <em>checker class</em>
628  that makes it possible to check whether a certain implementation of a
629  concept indeed provides all the required features.
630
631- Finally, They can serve as a skeleton of a new implementation of a concept.
632*/
633
634/**
635@defgroup graph_concepts Graph Structure Concepts
636@ingroup concept
637\brief Skeleton and concept checking classes for graph structures
638
639This group contains the skeletons and concept checking classes of LEMON's
640graph structures and helper classes used to implement these.
641*/
642
643/**
644@defgroup map_concepts Map Concepts
645@ingroup concept
646\brief Skeleton and concept checking classes for maps
647
648This group contains the skeletons and concept checking classes of maps.
649*/
650
651/**
652\anchor demoprograms
653
654@defgroup demos Demo Programs
655
656Some demo programs are listed here. Their full source codes can be found in
657the \c demo subdirectory of the source tree.
658
659In order to compile them, use the <tt>make demo</tt> or the
660<tt>make check</tt> commands.
661*/
662
663/**
664@defgroup tools Standalone Utility Applications
665
666Some utility applications are listed here.
667
668The standard compilation procedure (<tt>./configure;make</tt>) will compile
669them, as well.
670*/
671
672}
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