alpar@209: /* -*- mode: C++; indent-tabs-mode: nil; -*-
alpar@40:  *
alpar@209:  * This file is a part of LEMON, a generic C++ optimization library.
alpar@40:  *
alpar@440:  * Copyright (C) 2003-2009
alpar@40:  * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
alpar@40:  * (Egervary Research Group on Combinatorial Optimization, EGRES).
alpar@40:  *
alpar@40:  * Permission to use, modify and distribute this software is granted
alpar@40:  * provided that this copyright notice appears in all copies. For
alpar@40:  * precise terms see the accompanying LICENSE file.
alpar@40:  *
alpar@40:  * This software is provided "AS IS" with no warranty of any kind,
alpar@40:  * express or implied, and with no claim as to its suitability for any
alpar@40:  * purpose.
alpar@40:  *
alpar@40:  */
alpar@40: 
kpeter@406: namespace lemon {
kpeter@406: 
alpar@40: /**
alpar@40: @defgroup datas Data Structures
kpeter@550: This group contains the several data structures implemented in LEMON.
alpar@40: */
alpar@40: 
alpar@40: /**
alpar@40: @defgroup graphs Graph Structures
alpar@40: @ingroup datas
alpar@40: \brief Graph structures implemented in LEMON.
alpar@40: 
alpar@209: The implementation of combinatorial algorithms heavily relies on
alpar@209: efficient graph implementations. LEMON offers data structures which are
alpar@209: planned to be easily used in an experimental phase of implementation studies,
alpar@209: and thereafter the program code can be made efficient by small modifications.
alpar@40: 
alpar@40: The most efficient implementation of diverse applications require the
alpar@40: usage of different physical graph implementations. These differences
alpar@40: appear in the size of graph we require to handle, memory or time usage
alpar@40: limitations or in the set of operations through which the graph can be
alpar@40: accessed.  LEMON provides several physical graph structures to meet
alpar@40: the diverging requirements of the possible users.  In order to save on
alpar@40: running time or on memory usage, some structures may fail to provide
kpeter@83: some graph features like arc/edge or node deletion.
alpar@40: 
alpar@209: Alteration of standard containers need a very limited number of
alpar@209: operations, these together satisfy the everyday requirements.
alpar@209: In the case of graph structures, different operations are needed which do
alpar@209: not alter the physical graph, but gives another view. If some nodes or
kpeter@83: arcs have to be hidden or the reverse oriented graph have to be used, then
alpar@209: this is the case. It also may happen that in a flow implementation
alpar@209: the residual graph can be accessed by another algorithm, or a node-set
alpar@209: is to be shrunk for another algorithm.
alpar@209: LEMON also provides a variety of graphs for these requirements called
alpar@209: \ref graph_adaptors "graph adaptors". Adaptors cannot be used alone but only
alpar@209: in conjunction with other graph representations.
alpar@40: 
alpar@40: You are free to use the graph structure that fit your requirements
alpar@40: the best, most graph algorithms and auxiliary data structures can be used
kpeter@314: with any graph structure.
kpeter@314: 
kpeter@314: <b>See also:</b> \ref graph_concepts "Graph Structure Concepts".
alpar@40: */
alpar@40: 
alpar@40: /**
kpeter@451: @defgroup graph_adaptors Adaptor Classes for Graphs
deba@416: @ingroup graphs
kpeter@451: \brief Adaptor classes for digraphs and graphs
kpeter@451: 
kpeter@451: This group contains several useful adaptor classes for digraphs and graphs.
deba@416: 
deba@416: The main parts of LEMON are the different graph structures, generic
kpeter@451: graph algorithms, graph concepts, which couple them, and graph
deba@416: adaptors. While the previous notions are more or less clear, the
deba@416: latter one needs further explanation. Graph adaptors are graph classes
deba@416: which serve for considering graph structures in different ways.
deba@416: 
deba@416: A short example makes this much clearer.  Suppose that we have an
kpeter@451: instance \c g of a directed graph type, say ListDigraph and an algorithm
deba@416: \code
deba@416: template <typename Digraph>
deba@416: int algorithm(const Digraph&);
deba@416: \endcode
deba@416: is needed to run on the reverse oriented graph.  It may be expensive
deba@416: (in time or in memory usage) to copy \c g with the reversed
deba@416: arcs.  In this case, an adaptor class is used, which (according
kpeter@451: to LEMON \ref concepts::Digraph "digraph concepts") works as a digraph.
kpeter@451: The adaptor uses the original digraph structure and digraph operations when
kpeter@451: methods of the reversed oriented graph are called.  This means that the adaptor
kpeter@451: have minor memory usage, and do not perform sophisticated algorithmic
deba@416: actions.  The purpose of it is to give a tool for the cases when a
deba@416: graph have to be used in a specific alteration.  If this alteration is
kpeter@451: obtained by a usual construction like filtering the node or the arc set or
deba@416: considering a new orientation, then an adaptor is worthwhile to use.
deba@416: To come back to the reverse oriented graph, in this situation
deba@416: \code
deba@416: template<typename Digraph> class ReverseDigraph;
deba@416: \endcode
deba@416: template class can be used. The code looks as follows
deba@416: \code
deba@416: ListDigraph g;
kpeter@451: ReverseDigraph<ListDigraph> rg(g);
deba@416: int result = algorithm(rg);
deba@416: \endcode
kpeter@451: During running the algorithm, the original digraph \c g is untouched.
kpeter@451: This techniques give rise to an elegant code, and based on stable
deba@416: graph adaptors, complex algorithms can be implemented easily.
deba@416: 
kpeter@451: In flow, circulation and matching problems, the residual
deba@416: graph is of particular importance. Combining an adaptor implementing
kpeter@451: this with shortest path algorithms or minimum mean cycle algorithms,
deba@416: a range of weighted and cardinality optimization algorithms can be
deba@416: obtained. For other examples, the interested user is referred to the
deba@416: detailed documentation of particular adaptors.
deba@416: 
deba@416: The behavior of graph adaptors can be very different. Some of them keep
deba@416: capabilities of the original graph while in other cases this would be
kpeter@451: meaningless. This means that the concepts that they meet depend
kpeter@451: on the graph adaptor, and the wrapped graph.
kpeter@451: For example, if an arc of a reversed digraph is deleted, this is carried
kpeter@451: out by deleting the corresponding arc of the original digraph, thus the
kpeter@451: adaptor modifies the original digraph.
kpeter@451: However in case of a residual digraph, this operation has no sense.
deba@416: 
deba@416: Let us stand one more example here to simplify your work.
kpeter@451: ReverseDigraph has constructor
deba@416: \code
deba@416: ReverseDigraph(Digraph& digraph);
deba@416: \endcode
kpeter@451: This means that in a situation, when a <tt>const %ListDigraph&</tt>
deba@416: reference to a graph is given, then it have to be instantiated with
kpeter@451: <tt>Digraph=const %ListDigraph</tt>.
deba@416: \code
deba@416: int algorithm1(const ListDigraph& g) {
kpeter@451:   ReverseDigraph<const ListDigraph> rg(g);
deba@416:   return algorithm2(rg);
deba@416: }
deba@416: \endcode
deba@416: */
deba@416: 
deba@416: /**
kpeter@50: @defgroup semi_adaptors Semi-Adaptor Classes for Graphs
alpar@40: @ingroup graphs
alpar@40: \brief Graph types between real graphs and graph adaptors.
alpar@40: 
kpeter@550: This group contains some graph types between real graphs and graph adaptors.
alpar@209: These classes wrap graphs to give new functionality as the adaptors do it.
kpeter@50: On the other hand they are not light-weight structures as the adaptors.
alpar@40: */
alpar@40: 
alpar@40: /**
alpar@209: @defgroup maps Maps
alpar@40: @ingroup datas
kpeter@50: \brief Map structures implemented in LEMON.
alpar@40: 
kpeter@550: This group contains the map structures implemented in LEMON.
kpeter@50: 
kpeter@314: LEMON provides several special purpose maps and map adaptors that e.g. combine
alpar@40: new maps from existing ones.
kpeter@314: 
kpeter@314: <b>See also:</b> \ref map_concepts "Map Concepts".
alpar@40: */
alpar@40: 
alpar@40: /**
alpar@209: @defgroup graph_maps Graph Maps
alpar@40: @ingroup maps
kpeter@83: \brief Special graph-related maps.
alpar@40: 
kpeter@550: This group contains maps that are specifically designed to assign
kpeter@406: values to the nodes and arcs/edges of graphs.
kpeter@406: 
kpeter@406: If you are looking for the standard graph maps (\c NodeMap, \c ArcMap,
kpeter@406: \c EdgeMap), see the \ref graph_concepts "Graph Structure Concepts".
alpar@40: */
alpar@40: 
alpar@40: /**
alpar@40: \defgroup map_adaptors Map Adaptors
alpar@40: \ingroup maps
alpar@40: \brief Tools to create new maps from existing ones
alpar@40: 
kpeter@550: This group contains map adaptors that are used to create "implicit"
kpeter@50: maps from other maps.
alpar@40: 
kpeter@406: Most of them are \ref concepts::ReadMap "read-only maps".
kpeter@83: They can make arithmetic and logical operations between one or two maps
kpeter@83: (negation, shifting, addition, multiplication, logical 'and', 'or',
kpeter@83: 'not' etc.) or e.g. convert a map to another one of different Value type.
alpar@40: 
kpeter@50: The typical usage of this classes is passing implicit maps to
alpar@40: algorithms.  If a function type algorithm is called then the function
alpar@40: type map adaptors can be used comfortable. For example let's see the
kpeter@314: usage of map adaptors with the \c graphToEps() function.
alpar@40: \code
alpar@40:   Color nodeColor(int deg) {
alpar@40:     if (deg >= 2) {
alpar@40:       return Color(0.5, 0.0, 0.5);
alpar@40:     } else if (deg == 1) {
alpar@40:       return Color(1.0, 0.5, 1.0);
alpar@40:     } else {
alpar@40:       return Color(0.0, 0.0, 0.0);
alpar@40:     }
alpar@40:   }
alpar@209: 
kpeter@83:   Digraph::NodeMap<int> degree_map(graph);
alpar@209: 
kpeter@314:   graphToEps(graph, "graph.eps")
alpar@40:     .coords(coords).scaleToA4().undirected()
kpeter@83:     .nodeColors(composeMap(functorToMap(nodeColor), degree_map))
alpar@40:     .run();
alpar@209: \endcode
kpeter@83: The \c functorToMap() function makes an \c int to \c Color map from the
kpeter@314: \c nodeColor() function. The \c composeMap() compose the \c degree_map
kpeter@83: and the previously created map. The composed map is a proper function to
kpeter@83: get the color of each node.
alpar@40: 
alpar@40: The usage with class type algorithms is little bit harder. In this
alpar@40: case the function type map adaptors can not be used, because the
kpeter@50: function map adaptors give back temporary objects.
alpar@40: \code
kpeter@83:   Digraph graph;
kpeter@83: 
kpeter@83:   typedef Digraph::ArcMap<double> DoubleArcMap;
kpeter@83:   DoubleArcMap length(graph);
kpeter@83:   DoubleArcMap speed(graph);
kpeter@83: 
kpeter@83:   typedef DivMap<DoubleArcMap, DoubleArcMap> TimeMap;
alpar@40:   TimeMap time(length, speed);
alpar@209: 
kpeter@83:   Dijkstra<Digraph, TimeMap> dijkstra(graph, time);
alpar@40:   dijkstra.run(source, target);
alpar@40: \endcode
kpeter@83: We have a length map and a maximum speed map on the arcs of a digraph.
kpeter@83: The minimum time to pass the arc can be calculated as the division of
kpeter@83: the two maps which can be done implicitly with the \c DivMap template
alpar@40: class. We use the implicit minimum time map as the length map of the
alpar@40: \c Dijkstra algorithm.
alpar@40: */
alpar@40: 
alpar@40: /**
alpar@209: @defgroup matrices Matrices
alpar@40: @ingroup datas
kpeter@50: \brief Two dimensional data storages implemented in LEMON.
alpar@40: 
kpeter@550: This group contains two dimensional data storages implemented in LEMON.
alpar@40: */
alpar@40: 
alpar@40: /**
alpar@40: @defgroup paths Path Structures
alpar@40: @ingroup datas
kpeter@318: \brief %Path structures implemented in LEMON.
alpar@40: 
kpeter@550: This group contains the path structures implemented in LEMON.
alpar@40: 
kpeter@50: LEMON provides flexible data structures to work with paths.
kpeter@50: All of them have similar interfaces and they can be copied easily with
kpeter@50: assignment operators and copy constructors. This makes it easy and
alpar@40: efficient to have e.g. the Dijkstra algorithm to store its result in
alpar@40: any kind of path structure.
alpar@40: 
alpar@40: \sa lemon::concepts::Path
alpar@40: */
alpar@40: 
alpar@40: /**
alpar@40: @defgroup auxdat Auxiliary Data Structures
alpar@40: @ingroup datas
kpeter@50: \brief Auxiliary data structures implemented in LEMON.
alpar@40: 
kpeter@550: This group contains some data structures implemented in LEMON in
alpar@40: order to make it easier to implement combinatorial algorithms.
alpar@40: */
alpar@40: 
alpar@40: /**
alpar@40: @defgroup algs Algorithms
kpeter@550: \brief This group contains the several algorithms
alpar@40: implemented in LEMON.
alpar@40: 
kpeter@550: This group contains the several algorithms
alpar@40: implemented in LEMON.
alpar@40: */
alpar@40: 
alpar@40: /**
alpar@40: @defgroup search Graph Search
alpar@40: @ingroup algs
kpeter@50: \brief Common graph search algorithms.
alpar@40: 
kpeter@550: This group contains the common graph search algorithms, namely
kpeter@406: \e breadth-first \e search (BFS) and \e depth-first \e search (DFS).
alpar@40: */
alpar@40: 
alpar@40: /**
kpeter@314: @defgroup shortest_path Shortest Path Algorithms
alpar@40: @ingroup algs
kpeter@50: \brief Algorithms for finding shortest paths.
alpar@40: 
kpeter@550: This group contains the algorithms for finding shortest paths in digraphs.
kpeter@406: 
kpeter@406:  - \ref Dijkstra algorithm for finding shortest paths from a source node
kpeter@406:    when all arc lengths are non-negative.
kpeter@406:  - \ref BellmanFord "Bellman-Ford" algorithm for finding shortest paths
kpeter@406:    from a source node when arc lenghts can be either positive or negative,
kpeter@406:    but the digraph should not contain directed cycles with negative total
kpeter@406:    length.
kpeter@406:  - \ref FloydWarshall "Floyd-Warshall" and \ref Johnson "Johnson" algorithms
kpeter@406:    for solving the \e all-pairs \e shortest \e paths \e problem when arc
kpeter@406:    lenghts can be either positive or negative, but the digraph should
kpeter@406:    not contain directed cycles with negative total length.
kpeter@406:  - \ref Suurballe A successive shortest path algorithm for finding
kpeter@406:    arc-disjoint paths between two nodes having minimum total length.
alpar@40: */
alpar@40: 
alpar@209: /**
kpeter@314: @defgroup max_flow Maximum Flow Algorithms
alpar@209: @ingroup algs
kpeter@50: \brief Algorithms for finding maximum flows.
alpar@40: 
kpeter@550: This group contains the algorithms for finding maximum flows and
alpar@40: feasible circulations.
alpar@40: 
kpeter@406: The \e maximum \e flow \e problem is to find a flow of maximum value between
kpeter@406: a single source and a single target. Formally, there is a \f$G=(V,A)\f$
kpeter@601: digraph, a \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function and
kpeter@406: \f$s, t \in V\f$ source and target nodes.
kpeter@601: A maximum flow is an \f$f: A\rightarrow\mathbf{R}^+_0\f$ solution of the
kpeter@406: following optimization problem.
alpar@40: 
kpeter@601: \f[ \max\sum_{sv\in A} f(sv) - \sum_{vs\in A} f(vs) \f]
kpeter@601: \f[ \sum_{uv\in A} f(uv) = \sum_{vu\in A} f(vu)
kpeter@601:     \quad \forall u\in V\setminus\{s,t\} \f]
kpeter@601: \f[ 0 \leq f(uv) \leq cap(uv) \quad \forall uv\in A \f]
alpar@40: 
kpeter@50: LEMON contains several algorithms for solving maximum flow problems:
kpeter@406: - \ref EdmondsKarp Edmonds-Karp algorithm.
kpeter@406: - \ref Preflow Goldberg-Tarjan's preflow push-relabel algorithm.
kpeter@406: - \ref DinitzSleatorTarjan Dinitz's blocking flow algorithm with dynamic trees.
kpeter@406: - \ref GoldbergTarjan Preflow push-relabel algorithm with dynamic trees.
alpar@40: 
kpeter@406: In most cases the \ref Preflow "Preflow" algorithm provides the
kpeter@406: fastest method for computing a maximum flow. All implementations
kpeter@649: also provide functions to query the minimum cut, which is the dual
kpeter@649: problem of maximum flow.
kpeter@649: 
kpeter@649: \ref Circulation is a preflow push-relabel algorithm implemented directly 
kpeter@649: for finding feasible circulations, which is a somewhat different problem,
kpeter@649: but it is strongly related to maximum flow.
kpeter@649: For more information, see \ref Circulation.
alpar@40: */
alpar@40: 
alpar@40: /**
kpeter@314: @defgroup min_cost_flow Minimum Cost Flow Algorithms
alpar@40: @ingroup algs
alpar@40: 
kpeter@50: \brief Algorithms for finding minimum cost flows and circulations.
alpar@40: 
kpeter@601: This group contains the algorithms for finding minimum cost flows and
alpar@209: circulations.
kpeter@406: 
kpeter@406: The \e minimum \e cost \e flow \e problem is to find a feasible flow of
kpeter@406: minimum total cost from a set of supply nodes to a set of demand nodes
kpeter@601: in a network with capacity constraints (lower and upper bounds)
kpeter@601: and arc costs.
kpeter@636: Formally, let \f$G=(V,A)\f$ be a digraph, \f$lower: A\rightarrow\mathbf{Z}\f$,
kpeter@636: \f$upper: A\rightarrow\mathbf{Z}\cup\{+\infty\}\f$ denote the lower and
kpeter@601: upper bounds for the flow values on the arcs, for which
kpeter@636: \f$lower(uv) \leq upper(uv)\f$ must hold for all \f$uv\in A\f$,
kpeter@636: \f$cost: A\rightarrow\mathbf{Z}\f$ denotes the cost per unit flow
kpeter@636: on the arcs and \f$sup: V\rightarrow\mathbf{Z}\f$ denotes the
kpeter@601: signed supply values of the nodes.
kpeter@601: If \f$sup(u)>0\f$, then \f$u\f$ is a supply node with \f$sup(u)\f$
kpeter@601: supply, if \f$sup(u)<0\f$, then \f$u\f$ is a demand node with
kpeter@601: \f$-sup(u)\f$ demand.
kpeter@636: A minimum cost flow is an \f$f: A\rightarrow\mathbf{Z}\f$ solution
kpeter@601: of the following optimization problem.
kpeter@406: 
kpeter@601: \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f]
kpeter@601: \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \geq
kpeter@601:     sup(u) \quad \forall u\in V \f]
kpeter@601: \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f]
kpeter@406: 
kpeter@601: The sum of the supply values, i.e. \f$\sum_{u\in V} sup(u)\f$ must be
kpeter@601: zero or negative in order to have a feasible solution (since the sum
kpeter@601: of the expressions on the left-hand side of the inequalities is zero).
kpeter@601: It means that the total demand must be greater or equal to the total
kpeter@601: supply and all the supplies have to be carried out from the supply nodes,
kpeter@601: but there could be demands that are not satisfied.
kpeter@601: If \f$\sum_{u\in V} sup(u)\f$ is zero, then all the supply/demand
kpeter@601: constraints have to be satisfied with equality, i.e. all demands
kpeter@601: have to be satisfied and all supplies have to be used.
kpeter@601: 
kpeter@601: If you need the opposite inequalities in the supply/demand constraints
kpeter@601: (i.e. the total demand is less than the total supply and all the demands
kpeter@601: have to be satisfied while there could be supplies that are not used),
kpeter@601: then you could easily transform the problem to the above form by reversing
kpeter@601: the direction of the arcs and taking the negative of the supply values
kpeter@601: (e.g. using \ref ReverseDigraph and \ref NegMap adaptors).
kpeter@601: However \ref NetworkSimplex algorithm also supports this form directly
kpeter@601: for the sake of convenience.
kpeter@601: 
kpeter@601: A feasible solution for this problem can be found using \ref Circulation.
kpeter@601: 
kpeter@601: Note that the above formulation is actually more general than the usual
kpeter@601: definition of the minimum cost flow problem, in which strict equalities
kpeter@601: are required in the supply/demand contraints, i.e.
kpeter@601: 
kpeter@601: \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) =
kpeter@601:     sup(u) \quad \forall u\in V. \f]
kpeter@601: 
kpeter@601: However if the sum of the supply values is zero, then these two problems
kpeter@601: are equivalent. So if you need the equality form, you have to ensure this
kpeter@601: additional contraint for the algorithms.
kpeter@601: 
kpeter@601: The dual solution of the minimum cost flow problem is represented by node 
kpeter@601: potentials \f$\pi: V\rightarrow\mathbf{Z}\f$.
kpeter@636: An \f$f: A\rightarrow\mathbf{Z}\f$ feasible solution of the problem
kpeter@601: is optimal if and only if for some \f$\pi: V\rightarrow\mathbf{Z}\f$
kpeter@601: node potentials the following \e complementary \e slackness optimality
kpeter@601: conditions hold.
kpeter@601: 
kpeter@601:  - For all \f$uv\in A\f$ arcs:
kpeter@601:    - if \f$cost^\pi(uv)>0\f$, then \f$f(uv)=lower(uv)\f$;
kpeter@601:    - if \f$lower(uv)<f(uv)<upper(uv)\f$, then \f$cost^\pi(uv)=0\f$;
kpeter@601:    - if \f$cost^\pi(uv)<0\f$, then \f$f(uv)=upper(uv)\f$.
kpeter@636:  - For all \f$u\in V\f$ nodes:
kpeter@601:    - if \f$\sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \neq sup(u)\f$,
kpeter@601:      then \f$\pi(u)=0\f$.
kpeter@601:  
kpeter@601: Here \f$cost^\pi(uv)\f$ denotes the \e reduced \e cost of the arc
kpeter@636: \f$uv\in A\f$ with respect to the potential function \f$\pi\f$, i.e.
kpeter@601: \f[ cost^\pi(uv) = cost(uv) + \pi(u) - \pi(v).\f]
kpeter@601: 
kpeter@636: All algorithms provide dual solution (node potentials) as well,
kpeter@601: if an optimal flow is found.
kpeter@601: 
kpeter@601: LEMON contains several algorithms for solving minimum cost flow problems.
kpeter@601:  - \ref NetworkSimplex Primal Network Simplex algorithm with various
kpeter@601:    pivot strategies.
kpeter@601:  - \ref CostScaling Push-Relabel and Augment-Relabel algorithms based on
kpeter@601:    cost scaling.
kpeter@601:  - \ref CapacityScaling Successive Shortest %Path algorithm with optional
kpeter@406:    capacity scaling.
kpeter@601:  - \ref CancelAndTighten The Cancel and Tighten algorithm.
kpeter@601:  - \ref CycleCanceling Cycle-Canceling algorithms.
kpeter@601: 
kpeter@601: Most of these implementations support the general inequality form of the
kpeter@601: minimum cost flow problem, but CancelAndTighten and CycleCanceling
kpeter@601: only support the equality form due to the primal method they use.
kpeter@601: 
kpeter@601: In general NetworkSimplex is the most efficient implementation,
kpeter@601: but in special cases other algorithms could be faster.
kpeter@601: For example, if the total supply and/or capacities are rather small,
kpeter@601: CapacityScaling is usually the fastest algorithm (without effective scaling).
alpar@40: */
alpar@40: 
alpar@40: /**
kpeter@314: @defgroup min_cut Minimum Cut Algorithms
alpar@209: @ingroup algs
alpar@40: 
kpeter@50: \brief Algorithms for finding minimum cut in graphs.
alpar@40: 
kpeter@550: This group contains the algorithms for finding minimum cut in graphs.
alpar@40: 
kpeter@406: The \e minimum \e cut \e problem is to find a non-empty and non-complete
kpeter@406: \f$X\f$ subset of the nodes with minimum overall capacity on
kpeter@406: outgoing arcs. Formally, there is a \f$G=(V,A)\f$ digraph, a
kpeter@406: \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function. The minimum
kpeter@50: cut is the \f$X\f$ solution of the next optimization problem:
alpar@40: 
alpar@210: \f[ \min_{X \subset V, X\not\in \{\emptyset, V\}}
kpeter@406:     \sum_{uv\in A, u\in X, v\not\in X}cap(uv) \f]
alpar@40: 
kpeter@50: LEMON contains several algorithms related to minimum cut problems:
alpar@40: 
kpeter@406: - \ref HaoOrlin "Hao-Orlin algorithm" for calculating minimum cut
kpeter@406:   in directed graphs.
kpeter@406: - \ref NagamochiIbaraki "Nagamochi-Ibaraki algorithm" for
kpeter@406:   calculating minimum cut in undirected graphs.
kpeter@550: - \ref GomoryHu "Gomory-Hu tree computation" for calculating
kpeter@406:   all-pairs minimum cut in undirected graphs.
alpar@40: 
alpar@40: If you want to find minimum cut just between two distinict nodes,
kpeter@406: see the \ref max_flow "maximum flow problem".
alpar@40: */
alpar@40: 
alpar@40: /**
kpeter@578: @defgroup graph_properties Connectivity and Other Graph Properties
alpar@40: @ingroup algs
kpeter@50: \brief Algorithms for discovering the graph properties
alpar@40: 
kpeter@550: This group contains the algorithms for discovering the graph properties
kpeter@50: like connectivity, bipartiteness, euler property, simplicity etc.
alpar@40: 
alpar@40: \image html edge_biconnected_components.png
alpar@40: \image latex edge_biconnected_components.eps "bi-edge-connected components" width=\textwidth
alpar@40: */
alpar@40: 
alpar@40: /**
kpeter@314: @defgroup planar Planarity Embedding and Drawing
alpar@40: @ingroup algs
kpeter@50: \brief Algorithms for planarity checking, embedding and drawing
alpar@40: 
kpeter@550: This group contains the algorithms for planarity checking,
alpar@210: embedding and drawing.
alpar@40: 
alpar@40: \image html planar.png
alpar@40: \image latex planar.eps "Plane graph" width=\textwidth
alpar@40: */
alpar@40: 
alpar@40: /**
kpeter@314: @defgroup matching Matching Algorithms
alpar@40: @ingroup algs
kpeter@50: \brief Algorithms for finding matchings in graphs and bipartite graphs.
alpar@40: 
kpeter@582: This group contains the algorithms for calculating
alpar@40: matchings in graphs and bipartite graphs. The general matching problem is
kpeter@582: finding a subset of the edges for which each node has at most one incident
kpeter@582: edge.
alpar@209: 
alpar@40: There are several different algorithms for calculate matchings in
alpar@40: graphs.  The matching problems in bipartite graphs are generally
alpar@40: easier than in general graphs. The goal of the matching optimization
kpeter@406: can be finding maximum cardinality, maximum weight or minimum cost
alpar@40: matching. The search can be constrained to find perfect or
alpar@40: maximum cardinality matching.
alpar@40: 
kpeter@406: The matching algorithms implemented in LEMON:
kpeter@406: - \ref MaxBipartiteMatching Hopcroft-Karp augmenting path algorithm
kpeter@406:   for calculating maximum cardinality matching in bipartite graphs.
kpeter@406: - \ref PrBipartiteMatching Push-relabel algorithm
kpeter@406:   for calculating maximum cardinality matching in bipartite graphs.
kpeter@406: - \ref MaxWeightedBipartiteMatching
kpeter@406:   Successive shortest path algorithm for calculating maximum weighted
kpeter@406:   matching and maximum weighted bipartite matching in bipartite graphs.
kpeter@406: - \ref MinCostMaxBipartiteMatching
kpeter@406:   Successive shortest path algorithm for calculating minimum cost maximum
kpeter@406:   matching in bipartite graphs.
kpeter@406: - \ref MaxMatching Edmond's blossom shrinking algorithm for calculating
kpeter@406:   maximum cardinality matching in general graphs.
kpeter@406: - \ref MaxWeightedMatching Edmond's blossom shrinking algorithm for calculating
kpeter@406:   maximum weighted matching in general graphs.
kpeter@406: - \ref MaxWeightedPerfectMatching
kpeter@406:   Edmond's blossom shrinking algorithm for calculating maximum weighted
kpeter@406:   perfect matching in general graphs.
alpar@40: 
alpar@40: \image html bipartite_matching.png
alpar@40: \image latex bipartite_matching.eps "Bipartite Matching" width=\textwidth
alpar@40: */
alpar@40: 
alpar@40: /**
kpeter@314: @defgroup spantree Minimum Spanning Tree Algorithms
alpar@40: @ingroup algs
kpeter@649: \brief Algorithms for finding minimum cost spanning trees and arborescences.
alpar@40: 
kpeter@649: This group contains the algorithms for finding minimum cost spanning
kpeter@649: trees and arborescences.
alpar@40: */
alpar@40: 
alpar@40: /**
kpeter@314: @defgroup auxalg Auxiliary Algorithms
alpar@40: @ingroup algs
kpeter@50: \brief Auxiliary algorithms implemented in LEMON.
alpar@40: 
kpeter@550: This group contains some algorithms implemented in LEMON
kpeter@50: in order to make it easier to implement complex algorithms.
alpar@40: */
alpar@40: 
alpar@40: /**
kpeter@314: @defgroup approx Approximation Algorithms
kpeter@314: @ingroup algs
kpeter@50: \brief Approximation algorithms.
alpar@40: 
kpeter@550: This group contains the approximation and heuristic algorithms
kpeter@50: implemented in LEMON.
alpar@40: */
alpar@40: 
alpar@40: /**
alpar@40: @defgroup gen_opt_group General Optimization Tools
kpeter@550: \brief This group contains some general optimization frameworks
alpar@40: implemented in LEMON.
alpar@40: 
kpeter@550: This group contains some general optimization frameworks
alpar@40: implemented in LEMON.
alpar@40: */
alpar@40: 
alpar@40: /**
kpeter@314: @defgroup lp_group Lp and Mip Solvers
alpar@40: @ingroup gen_opt_group
alpar@40: \brief Lp and Mip solver interfaces for LEMON.
alpar@40: 
kpeter@550: This group contains Lp and Mip solver interfaces for LEMON. The
alpar@40: various LP solvers could be used in the same manner with this
alpar@40: interface.
alpar@40: */
alpar@40: 
alpar@209: /**
kpeter@314: @defgroup lp_utils Tools for Lp and Mip Solvers
alpar@40: @ingroup lp_group
kpeter@50: \brief Helper tools to the Lp and Mip solvers.
alpar@40: 
alpar@40: This group adds some helper tools to general optimization framework
alpar@40: implemented in LEMON.
alpar@40: */
alpar@40: 
alpar@40: /**
alpar@40: @defgroup metah Metaheuristics
alpar@40: @ingroup gen_opt_group
alpar@40: \brief Metaheuristics for LEMON library.
alpar@40: 
kpeter@550: This group contains some metaheuristic optimization tools.
alpar@40: */
alpar@40: 
alpar@40: /**
alpar@209: @defgroup utils Tools and Utilities
kpeter@50: \brief Tools and utilities for programming in LEMON
alpar@40: 
kpeter@50: Tools and utilities for programming in LEMON.
alpar@40: */
alpar@40: 
alpar@40: /**
alpar@40: @defgroup gutils Basic Graph Utilities
alpar@40: @ingroup utils
kpeter@50: \brief Simple basic graph utilities.
alpar@40: 
kpeter@550: This group contains some simple basic graph utilities.
alpar@40: */
alpar@40: 
alpar@40: /**
alpar@40: @defgroup misc Miscellaneous Tools
alpar@40: @ingroup utils
kpeter@50: \brief Tools for development, debugging and testing.
kpeter@50: 
kpeter@550: This group contains several useful tools for development,
alpar@40: debugging and testing.
alpar@40: */
alpar@40: 
alpar@40: /**
kpeter@314: @defgroup timecount Time Measuring and Counting
alpar@40: @ingroup misc
kpeter@50: \brief Simple tools for measuring the performance of algorithms.
kpeter@50: 
kpeter@550: This group contains simple tools for measuring the performance
alpar@40: of algorithms.
alpar@40: */
alpar@40: 
alpar@40: /**
alpar@40: @defgroup exceptions Exceptions
alpar@40: @ingroup utils
kpeter@50: \brief Exceptions defined in LEMON.
kpeter@50: 
kpeter@550: This group contains the exceptions defined in LEMON.
alpar@40: */
alpar@40: 
alpar@40: /**
alpar@40: @defgroup io_group Input-Output
kpeter@50: \brief Graph Input-Output methods
alpar@40: 
kpeter@550: This group contains the tools for importing and exporting graphs
kpeter@314: and graph related data. Now it supports the \ref lgf-format
kpeter@314: "LEMON Graph Format", the \c DIMACS format and the encapsulated
kpeter@314: postscript (EPS) format.
alpar@40: */
alpar@40: 
alpar@40: /**
kpeter@351: @defgroup lemon_io LEMON Graph Format
alpar@40: @ingroup io_group
kpeter@314: \brief Reading and writing LEMON Graph Format.
alpar@40: 
kpeter@550: This group contains methods for reading and writing
ladanyi@236: \ref lgf-format "LEMON Graph Format".
alpar@40: */
alpar@40: 
alpar@40: /**
kpeter@314: @defgroup eps_io Postscript Exporting
alpar@40: @ingroup io_group
alpar@40: \brief General \c EPS drawer and graph exporter
alpar@40: 
kpeter@550: This group contains general \c EPS drawing methods and special
alpar@209: graph exporting tools.
alpar@40: */
alpar@40: 
alpar@40: /**
kpeter@388: @defgroup dimacs_group DIMACS format
kpeter@388: @ingroup io_group
kpeter@388: \brief Read and write files in DIMACS format
kpeter@388: 
kpeter@388: Tools to read a digraph from or write it to a file in DIMACS format data.
kpeter@388: */
kpeter@388: 
kpeter@388: /**
kpeter@351: @defgroup nauty_group NAUTY Format
kpeter@351: @ingroup io_group
kpeter@351: \brief Read \e Nauty format
kpeter@388: 
kpeter@351: Tool to read graphs from \e Nauty format data.
kpeter@351: */
kpeter@351: 
kpeter@351: /**
alpar@40: @defgroup concept Concepts
alpar@40: \brief Skeleton classes and concept checking classes
alpar@40: 
kpeter@550: This group contains the data/algorithm skeletons and concept checking
alpar@40: classes implemented in LEMON.
alpar@40: 
alpar@40: The purpose of the classes in this group is fourfold.
alpar@209: 
kpeter@318: - These classes contain the documentations of the %concepts. In order
alpar@40:   to avoid document multiplications, an implementation of a concept
alpar@40:   simply refers to the corresponding concept class.
alpar@40: 
alpar@40: - These classes declare every functions, <tt>typedef</tt>s etc. an
kpeter@318:   implementation of the %concepts should provide, however completely
alpar@40:   without implementations and real data structures behind the
alpar@40:   interface. On the other hand they should provide nothing else. All
alpar@40:   the algorithms working on a data structure meeting a certain concept
alpar@40:   should compile with these classes. (Though it will not run properly,
alpar@40:   of course.) In this way it is easily to check if an algorithm
alpar@40:   doesn't use any extra feature of a certain implementation.
alpar@40: 
alpar@40: - The concept descriptor classes also provide a <em>checker class</em>
kpeter@50:   that makes it possible to check whether a certain implementation of a
alpar@40:   concept indeed provides all the required features.
alpar@40: 
alpar@40: - Finally, They can serve as a skeleton of a new implementation of a concept.
alpar@40: */
alpar@40: 
alpar@40: /**
alpar@40: @defgroup graph_concepts Graph Structure Concepts
alpar@40: @ingroup concept
alpar@40: \brief Skeleton and concept checking classes for graph structures
alpar@40: 
kpeter@550: This group contains the skeletons and concept checking classes of LEMON's
alpar@40: graph structures and helper classes used to implement these.
alpar@40: */
alpar@40: 
kpeter@314: /**
kpeter@314: @defgroup map_concepts Map Concepts
kpeter@314: @ingroup concept
kpeter@314: \brief Skeleton and concept checking classes for maps
kpeter@314: 
kpeter@550: This group contains the skeletons and concept checking classes of maps.
alpar@40: */
alpar@40: 
alpar@40: /**
alpar@40: \anchor demoprograms
alpar@40: 
kpeter@406: @defgroup demos Demo Programs
alpar@40: 
alpar@40: Some demo programs are listed here. Their full source codes can be found in
alpar@40: the \c demo subdirectory of the source tree.
alpar@40: 
ladanyi@556: In order to compile them, use the <tt>make demo</tt> or the
ladanyi@556: <tt>make check</tt> commands.
alpar@40: */
alpar@40: 
alpar@40: /**
kpeter@406: @defgroup tools Standalone Utility Applications
alpar@40: 
alpar@209: Some utility applications are listed here.
alpar@40: 
alpar@40: The standard compilation procedure (<tt>./configure;make</tt>) will compile
alpar@209: them, as well.
alpar@40: */
alpar@40: 
kpeter@406: }