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