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@…>, 12 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
[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 */
[422]19namespace lemon {
22@defgroup datas Data Structures
[606]23This group contains the several data structures implemented in LEMON.
27@defgroup graphs Graph Structures
28@ingroup datas
29\brief Graph structures implemented in LEMON.
[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.
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.
[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.
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.
61<b>See also:</b> \ref graph_concepts "Graph Structure Concepts".
[474]65@defgroup graph_adaptors Adaptor Classes for Graphs
[432]66@ingroup graphs
[474]67\brief Adaptor classes for digraphs and graphs
69This group contains several useful adaptor classes for digraphs and graphs.
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.
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
80template <typename Digraph>
81int algorithm(const Digraph&);
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
96template<typename Digraph> class ReverseDigraph;
98template class can be used. The code looks as follows
100ListDigraph g;
[474]101ReverseDigraph<ListDigraph> rg(g);
[432]102int result = algorithm(rg);
[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.
[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.
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.
124Let us stand one more example here to simplify your work.
[474]125ReverseDigraph has constructor
127ReverseDigraph(Digraph& digraph);
[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>.
133int algorithm1(const ListDigraph& g) {
[474]134  ReverseDigraph<const ListDigraph> rg(g);
[432]135  return algorithm2(rg);
[209]141@defgroup maps Maps
[40]142@ingroup datas
[50]143\brief Map structures implemented in LEMON.
[606]145This group contains the map structures implemented in LEMON.
[314]147LEMON provides several special purpose maps and map adaptors that e.g. combine
[40]148new maps from existing ones.
150<b>See also:</b> \ref map_concepts "Map Concepts".
[209]154@defgroup graph_maps Graph Maps
[40]155@ingroup maps
[83]156\brief Special graph-related maps.
[606]158This group contains maps that are specifically designed to assign
[422]159values to the nodes and arcs/edges of graphs.
161If you are looking for the standard graph maps (\c NodeMap, \c ArcMap,
162\c EdgeMap), see the \ref graph_concepts "Graph Structure Concepts".
166\defgroup map_adaptors Map Adaptors
167\ingroup maps
168\brief Tools to create new maps from existing ones
[606]170This group contains map adaptors that are used to create "implicit"
[50]171maps from other maps.
[422]173Most of them are \ref concepts::ReadMap "read-only maps".
[83]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.
[50]178The typical usage of this classes is passing implicit maps to
[40]179algorithms.  If a function type algorithm is called then the function
180type map adaptors can be used comfortable. For example let's see the
[314]181usage of map adaptors with the \c graphToEps() function.
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  }
[83]193  Digraph::NodeMap<int> degree_map(graph);
[314]195  graphToEps(graph, "graph.eps")
[40]196    .coords(coords).scaleToA4().undirected()
[83]197    .nodeColors(composeMap(functorToMap(nodeColor), degree_map))
[40]198    .run();
[83]200The \c functorToMap() function makes an \c int to \c Color map from the
[314]201\c nodeColor() function. The \c composeMap() compose the \c degree_map
[83]202and the previously created map. The composed map is a proper function to
203get the color of each node.
205The usage with class type algorithms is little bit harder. In this
206case the function type map adaptors can not be used, because the
[50]207function map adaptors give back temporary objects.
[83]209  Digraph graph;
211  typedef Digraph::ArcMap<double> DoubleArcMap;
212  DoubleArcMap length(graph);
213  DoubleArcMap speed(graph);
215  typedef DivMap<DoubleArcMap, DoubleArcMap> TimeMap;
[40]216  TimeMap time(length, speed);
[83]218  Dijkstra<Digraph, TimeMap> dijkstra(graph, time);
[40]219, target);
[83]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
[40]224class. We use the implicit minimum time map as the length map of the
225\c Dijkstra algorithm.
[209]229@defgroup matrices Matrices
[40]230@ingroup datas
[50]231\brief Two dimensional data storages implemented in LEMON.
[606]233This group contains two dimensional data storages implemented in LEMON.
237@defgroup paths Path Structures
238@ingroup datas
[318]239\brief %Path structures implemented in LEMON.
[606]241This group contains the path structures implemented in LEMON.
[50]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
[40]246efficient to have e.g. the Dijkstra algorithm to store its result in
247any kind of path structure.
249\sa lemon::concepts::Path
253@defgroup auxdat Auxiliary Data Structures
254@ingroup datas
[50]255\brief Auxiliary data structures implemented in LEMON.
[606]257This group contains some data structures implemented in LEMON in
[40]258order to make it easier to implement combinatorial algorithms.
262@defgroup algs Algorithms
[606]263\brief This group contains the several algorithms
[40]264implemented in LEMON.
[606]266This group contains the several algorithms
[40]267implemented in LEMON.
271@defgroup search Graph Search
272@ingroup algs
[50]273\brief Common graph search algorithms.
[606]275This group contains the common graph search algorithms, namely
[422]276\e breadth-first \e search (BFS) and \e depth-first \e search (DFS).
[314]280@defgroup shortest_path Shortest Path Algorithms
[40]281@ingroup algs
[50]282\brief Algorithms for finding shortest paths.
[606]284This group contains the algorithms for finding shortest paths in digraphs.
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.
[314]301@defgroup max_flow Maximum Flow Algorithms
[209]302@ingroup algs
[50]303\brief Algorithms for finding maximum flows.
[606]305This group contains the algorithms for finding maximum flows and
[40]306feasible circulations.
[422]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$G=(V,A)\f$
[656]310digraph, a \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function and
[422]311\f$s, t \in V\f$ source and target nodes.
[656]312A maximum flow is an \f$f: A\rightarrow\mathbf{R}^+_0\f$ solution of the
[422]313following optimization problem.
[656]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]
[50]320LEMON contains several algorithms for solving maximum flow problems:
[422]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.
[422]326In most cases the \ref Preflow "Preflow" algorithm provides the
327fastest method for computing a maximum flow. All implementations
[698]328also provide functions to query the minimum cut, which is the dual
329problem of maximum flow.
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.
[710]338@defgroup min_cost_flow_algs Minimum Cost Flow Algorithms
[40]339@ingroup algs
[50]341\brief Algorithms for finding minimum cost flows and circulations.
[656]343This group contains the algorithms for finding minimum cost flows and
[710]344circulations. For more information about this problem and its dual
345solution see \ref min_cost_flow "Minimum Cost Flow Problem".
[710]347LEMON contains several algorithms for this problem.
[656]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
[422]353   capacity scaling.
[656]354 - \ref CancelAndTighten The Cancel and Tighten algorithm.
355 - \ref CycleCanceling Cycle-Canceling algorithms.
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).
[314]364@defgroup min_cut Minimum Cut Algorithms
[209]365@ingroup algs
[50]367\brief Algorithms for finding minimum cut in graphs.
[606]369This group contains the algorithms for finding minimum cut in graphs.
[422]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
[50]375cut is the \f$X\f$ solution of the next optimization problem:
[210]377\f[ \min_{X \subset V, X\not\in \{\emptyset, V\}}
[422]378    \sum_{uv\in A, u\in X, v\not\in X}cap(uv) \f]
[50]380LEMON contains several algorithms related to minimum cut problems:
[422]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.
[606]386- \ref GomoryHu "Gomory-Hu tree computation" for calculating
[422]387  all-pairs minimum cut in undirected graphs.
389If you want to find minimum cut just between two distinict nodes,
[422]390see the \ref max_flow "maximum flow problem".
[633]394@defgroup graph_properties Connectivity and Other Graph Properties
[40]395@ingroup algs
[50]396\brief Algorithms for discovering the graph properties
[606]398This group contains the algorithms for discovering the graph properties
[50]399like connectivity, bipartiteness, euler property, simplicity etc.
401\image html edge_biconnected_components.png
402\image latex edge_biconnected_components.eps "bi-edge-connected components" width=\textwidth
[314]406@defgroup planar Planarity Embedding and Drawing
[40]407@ingroup algs
[50]408\brief Algorithms for planarity checking, embedding and drawing
[606]410This group contains the algorithms for planarity checking,
[210]411embedding and drawing.
413\image html planar.png
414\image latex planar.eps "Plane graph" width=\textwidth
[314]418@defgroup matching Matching Algorithms
[40]419@ingroup algs
[50]420\brief Algorithms for finding matchings in graphs and bipartite graphs.
[637]422This group contains the algorithms for calculating
[40]423matchings in graphs and bipartite graphs. The general matching problem is
[637]424finding a subset of the edges for which each node has at most one incident
[40]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
[422]430can be finding maximum cardinality, maximum weight or minimum cost
[40]431matching. The search can be constrained to find perfect or
432maximum cardinality matching.
[422]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.
453\image html bipartite_matching.png
454\image latex bipartite_matching.eps "Bipartite Matching" width=\textwidth
[314]458@defgroup spantree Minimum Spanning Tree Algorithms
[40]459@ingroup algs
[698]460\brief Algorithms for finding minimum cost spanning trees and arborescences.
[698]462This group contains the algorithms for finding minimum cost spanning
463trees and arborescences.
[314]467@defgroup auxalg Auxiliary Algorithms
[40]468@ingroup algs
[50]469\brief Auxiliary algorithms implemented in LEMON.
[606]471This group contains some algorithms implemented in LEMON
[50]472in order to make it easier to implement complex algorithms.
[314]476@defgroup approx Approximation Algorithms
477@ingroup algs
[50]478\brief Approximation algorithms.
[606]480This group contains the approximation and heuristic algorithms
[50]481implemented in LEMON.
485@defgroup gen_opt_group General Optimization Tools
[606]486\brief This group contains some general optimization frameworks
[40]487implemented in LEMON.
[606]489This group contains some general optimization frameworks
[40]490implemented in LEMON.
[314]494@defgroup lp_group Lp and Mip Solvers
[40]495@ingroup gen_opt_group
496\brief Lp and Mip solver interfaces for LEMON.
[606]498This group contains Lp and Mip solver interfaces for LEMON. The
[40]499various LP solvers could be used in the same manner with this
[314]504@defgroup lp_utils Tools for Lp and Mip Solvers
[40]505@ingroup lp_group
[50]506\brief Helper tools to the Lp and Mip solvers.
508This group adds some helper tools to general optimization framework
509implemented in LEMON.
513@defgroup metah Metaheuristics
514@ingroup gen_opt_group
515\brief Metaheuristics for LEMON library.
[606]517This group contains some metaheuristic optimization tools.
[209]521@defgroup utils Tools and Utilities
[50]522\brief Tools and utilities for programming in LEMON
[50]524Tools and utilities for programming in LEMON.
528@defgroup gutils Basic Graph Utilities
529@ingroup utils
[50]530\brief Simple basic graph utilities.
[606]532This group contains some simple basic graph utilities.
536@defgroup misc Miscellaneous Tools
537@ingroup utils
[50]538\brief Tools for development, debugging and testing.
[606]540This group contains several useful tools for development,
[40]541debugging and testing.
[314]545@defgroup timecount Time Measuring and Counting
[40]546@ingroup misc
[50]547\brief Simple tools for measuring the performance of algorithms.
[606]549This group contains simple tools for measuring the performance
[40]550of algorithms.
554@defgroup exceptions Exceptions
555@ingroup utils
[50]556\brief Exceptions defined in LEMON.
[606]558This group contains the exceptions defined in LEMON.
562@defgroup io_group Input-Output
[50]563\brief Graph Input-Output methods
[606]565This group contains the tools for importing and exporting graphs
[314]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.
[363]572@defgroup lemon_io LEMON Graph Format
[40]573@ingroup io_group
[314]574\brief Reading and writing LEMON Graph Format.
[606]576This group contains methods for reading and writing
[236]577\ref lgf-format "LEMON Graph Format".
[314]581@defgroup eps_io Postscript Exporting
[40]582@ingroup io_group
583\brief General \c EPS drawer and graph exporter
[606]585This group contains general \c EPS drawing methods and special
[209]586graph exporting tools.
[403]590@defgroup dimacs_group DIMACS format
591@ingroup io_group
592\brief Read and write files in DIMACS format
594Tools to read a digraph from or write it to a file in DIMACS format data.
[363]598@defgroup nauty_group NAUTY Format
599@ingroup io_group
600\brief Read \e Nauty format
[363]602Tool to read graphs from \e Nauty format data.
[40]606@defgroup concept Concepts
607\brief Skeleton classes and concept checking classes
[606]609This group contains the data/algorithm skeletons and concept checking
[40]610classes implemented in LEMON.
612The purpose of the classes in this group is fourfold.
[318]614- These classes contain the documentations of the %concepts. In order
[40]615  to avoid document multiplications, an implementation of a concept
616  simply refers to the corresponding concept class.
618- These classes declare every functions, <tt>typedef</tt>s etc. an
[318]619  implementation of the %concepts should provide, however completely
[40]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.
627- The concept descriptor classes also provide a <em>checker class</em>
[50]628  that makes it possible to check whether a certain implementation of a
[40]629  concept indeed provides all the required features.
631- Finally, They can serve as a skeleton of a new implementation of a concept.
635@defgroup graph_concepts Graph Structure Concepts
636@ingroup concept
637\brief Skeleton and concept checking classes for graph structures
[606]639This group contains the skeletons and concept checking classes of LEMON's
[40]640graph structures and helper classes used to implement these.
644@defgroup map_concepts Map Concepts
645@ingroup concept
646\brief Skeleton and concept checking classes for maps
[606]648This group contains the skeletons and concept checking classes of maps.
652\anchor demoprograms
[422]654@defgroup demos Demo Programs
656Some demo programs are listed here. Their full source codes can be found in
657the \c demo subdirectory of the source tree.
[611]659In order to compile them, use the <tt>make demo</tt> or the
660<tt>make check</tt> commands.
[422]664@defgroup tools Standalone Utility Applications
[209]666Some utility applications are listed here.
668The standard compilation procedure (<tt>./configure;make</tt>) will compile
[209]669them, as well.
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