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@1092: * Copyright (C) 2003-2013 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@559: 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: See also: \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 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 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 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: kpeter@1050: Since the adaptor classes conform to the \ref graph_concepts "graph concepts", kpeter@1050: an adaptor can even be applied to another one. kpeter@1050: The following image illustrates a situation when a \ref SubDigraph adaptor kpeter@1050: is applied on a digraph and \ref Undirector is applied on the subgraph. kpeter@1050: kpeter@1050: \image html adaptors2.png kpeter@1050: \image latex adaptors2.eps "Using graph adaptors" width=\textwidth kpeter@1050: 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 const %ListDigraph& deba@416: reference to a graph is given, then it have to be instantiated with kpeter@451: Digraph=const %ListDigraph. deba@416: \code deba@416: int algorithm1(const ListDigraph& g) { kpeter@451: ReverseDigraph rg(g); deba@416: return algorithm2(rg); deba@416: } deba@416: \endcode deba@416: */ deba@416: deba@416: /** alpar@209: @defgroup maps Maps alpar@40: @ingroup datas kpeter@50: \brief Map structures implemented in LEMON. alpar@40: kpeter@559: 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: See also: \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@559: 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@559: 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 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 DoubleArcMap; kpeter@83: DoubleArcMap length(graph); kpeter@83: DoubleArcMap speed(graph); kpeter@83: kpeter@83: typedef DivMap TimeMap; alpar@40: TimeMap time(length, speed); alpar@209: kpeter@83: Dijkstra 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@40: @defgroup paths Path Structures alpar@40: @ingroup datas kpeter@318: \brief %Path structures implemented in LEMON. alpar@40: kpeter@559: 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: kpeter@710: \sa \ref concepts::Path "Path concept" kpeter@710: */ kpeter@710: kpeter@710: /** kpeter@710: @defgroup heaps Heap Structures kpeter@710: @ingroup datas kpeter@710: \brief %Heap structures implemented in LEMON. kpeter@710: kpeter@710: This group contains the heap structures implemented in LEMON. kpeter@710: kpeter@710: LEMON provides several heap classes. They are efficient implementations kpeter@710: of the abstract data type \e priority \e queue. They store items with kpeter@710: specified values called \e priorities in such a way that finding and kpeter@710: removing the item with minimum priority are efficient. kpeter@710: The basic operations are adding and erasing items, changing the priority kpeter@710: of an item, etc. kpeter@710: kpeter@710: Heaps are crucial in several algorithms, such as Dijkstra and Prim. kpeter@710: The heap implementations have the same interface, thus any of them can be kpeter@710: used easily in such algorithms. kpeter@710: kpeter@710: \sa \ref concepts::Heap "Heap concept" kpeter@710: */ kpeter@710: kpeter@710: /** alpar@40: @defgroup auxdat Auxiliary Data Structures alpar@40: @ingroup datas kpeter@50: \brief Auxiliary data structures implemented in LEMON. alpar@40: kpeter@559: 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: /** kpeter@714: @defgroup geomdat Geometric Data Structures kpeter@714: @ingroup auxdat kpeter@714: \brief Geometric data structures implemented in LEMON. kpeter@714: kpeter@714: This group contains geometric data structures implemented in LEMON. kpeter@714: kpeter@714: - \ref lemon::dim2::Point "dim2::Point" implements a two dimensional kpeter@714: vector with the usual operations. kpeter@714: - \ref lemon::dim2::Box "dim2::Box" can be used to determine the kpeter@714: rectangular bounding box of a set of \ref lemon::dim2::Point kpeter@714: "dim2::Point"'s. kpeter@714: */ kpeter@714: kpeter@714: /** kpeter@714: @defgroup matrices Matrices kpeter@714: @ingroup auxdat kpeter@714: \brief Two dimensional data storages implemented in LEMON. kpeter@714: kpeter@714: This group contains two dimensional data storages implemented in LEMON. kpeter@714: */ kpeter@714: kpeter@714: /** alpar@40: @defgroup algs Algorithms kpeter@559: \brief This group contains the several algorithms alpar@40: implemented in LEMON. alpar@40: kpeter@559: 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@559: This group contains the common graph search algorithms, namely kpeter@755: \e breadth-first \e search (BFS) and \e depth-first \e search (DFS) alpar@1053: \cite clrs01algorithms. 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@755: This group contains the algorithms for finding shortest paths in digraphs alpar@1053: \cite clrs01algorithms. 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@714: @defgroup spantree Minimum Spanning Tree Algorithms kpeter@714: @ingroup algs kpeter@714: \brief Algorithms for finding minimum cost spanning trees and arborescences. kpeter@714: kpeter@714: This group contains the algorithms for finding minimum cost spanning alpar@1053: trees and arborescences \cite clrs01algorithms. kpeter@714: */ kpeter@714: kpeter@714: /** kpeter@314: @defgroup max_flow Maximum Flow Algorithms alpar@209: @ingroup algs kpeter@50: \brief Algorithms for finding maximum flows. alpar@40: kpeter@559: This group contains the algorithms for finding maximum flows and alpar@1053: feasible circulations \cite clrs01algorithms, \cite amo93networkflows. 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@609: 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@609: A maximum flow is an \f$f: A\rightarrow\mathbf{R}^+_0\f$ solution of the kpeter@406: following optimization problem. alpar@40: kpeter@609: \f[ \max\sum_{sv\in A} f(sv) - \sum_{vs\in A} f(vs) \f] kpeter@609: \f[ \sum_{uv\in A} f(uv) = \sum_{vu\in A} f(vu) kpeter@609: \quad \forall u\in V\setminus\{s,t\} \f] kpeter@609: \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@755: - \ref EdmondsKarp Edmonds-Karp algorithm alpar@1053: \cite edmondskarp72theoretical. kpeter@755: - \ref Preflow Goldberg-Tarjan's preflow push-relabel algorithm alpar@1053: \cite goldberg88newapproach. kpeter@755: - \ref DinitzSleatorTarjan Dinitz's blocking flow algorithm with dynamic trees alpar@1053: \cite dinic70algorithm, \cite sleator83dynamic. kpeter@755: - \ref GoldbergTarjan !Preflow push-relabel algorithm with dynamic trees alpar@1053: \cite goldberg88newapproach, \cite sleator83dynamic. alpar@40: kpeter@755: In most cases the \ref Preflow algorithm provides the kpeter@406: fastest method for computing a maximum flow. All implementations kpeter@651: also provide functions to query the minimum cut, which is the dual kpeter@651: problem of maximum flow. kpeter@651: deba@869: \ref Circulation is a preflow push-relabel algorithm implemented directly kpeter@651: for finding feasible circulations, which is a somewhat different problem, kpeter@651: but it is strongly related to maximum flow. kpeter@651: For more information, see \ref Circulation. alpar@40: */ alpar@40: alpar@40: /** kpeter@663: @defgroup min_cost_flow_algs Minimum Cost Flow Algorithms alpar@40: @ingroup algs alpar@40: kpeter@50: \brief Algorithms for finding minimum cost flows and circulations. alpar@40: kpeter@609: This group contains the algorithms for finding minimum cost flows and alpar@1053: circulations \cite amo93networkflows. For more information about this kpeter@1049: problem and its dual solution, see: \ref min_cost_flow kpeter@755: "Minimum Cost Flow Problem". kpeter@406: kpeter@663: LEMON contains several algorithms for this problem. kpeter@609: - \ref NetworkSimplex Primal Network Simplex algorithm with various alpar@1053: pivot strategies \cite dantzig63linearprog, \cite kellyoneill91netsimplex. kpeter@813: - \ref CostScaling Cost Scaling algorithm based on push/augment and alpar@1053: relabel operations \cite goldberg90approximation, \cite goldberg97efficient, alpar@1053: \cite bunnagel98efficient. kpeter@813: - \ref CapacityScaling Capacity Scaling algorithm based on the successive alpar@1053: shortest path method \cite edmondskarp72theoretical. kpeter@813: - \ref CycleCanceling Cycle-Canceling algorithms, two of which are alpar@1053: strongly polynomial \cite klein67primal, \cite goldberg89cyclecanceling. kpeter@609: kpeter@919: In general, \ref NetworkSimplex and \ref CostScaling are the most efficient kpeter@1003: implementations. kpeter@1003: \ref NetworkSimplex is usually the fastest on relatively small graphs (up to kpeter@1003: several thousands of nodes) and on dense graphs, while \ref CostScaling is kpeter@1003: typically more efficient on large graphs (e.g. hundreds of thousands of kpeter@1003: nodes or above), especially if they are sparse. kpeter@1003: However, other algorithms could be faster in special cases. kpeter@609: For example, if the total supply and/or capacities are rather small, alpar@1093: \ref CapacityScaling is usually the fastest algorithm alpar@1093: (without effective scaling). kpeter@1002: kpeter@1002: These classes are intended to be used with integer-valued input data kpeter@1002: (capacities, supply values, and costs), except for \ref CapacityScaling, kpeter@1002: which is capable of handling real-valued arc costs (other numerical kpeter@1002: data are required to be integer). kpeter@1051: alpar@1092: For more details about these implementations and for a comprehensive alpar@1053: experimental study, see the paper \cite KiralyKovacs12MCF. kpeter@1051: It also compares these codes to other publicly available kpeter@1051: minimum cost flow solvers. 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@559: 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@713: \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@559: - \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@768: @defgroup min_mean_cycle Minimum Mean Cycle Algorithms alpar@40: @ingroup algs kpeter@768: \brief Algorithms for finding minimum mean cycles. alpar@40: kpeter@771: This group contains the algorithms for finding minimum mean cycles alpar@1053: \cite amo93networkflows, \cite karp78characterization. alpar@40: kpeter@768: The \e minimum \e mean \e cycle \e problem is to find a directed cycle kpeter@768: of minimum mean length (cost) in a digraph. kpeter@768: The mean length of a cycle is the average length of its arcs, i.e. the kpeter@768: ratio between the total length of the cycle and the number of arcs on it. alpar@40: kpeter@768: This problem has an important connection to \e conservative \e length kpeter@768: \e functions, too. A length function on the arcs of a digraph is called kpeter@768: conservative if and only if there is no directed cycle of negative total kpeter@768: length. For an arbitrary length function, the negative of the minimum kpeter@768: cycle mean is the smallest \f$\epsilon\f$ value so that increasing the kpeter@768: arc lengths uniformly by \f$\epsilon\f$ results in a conservative length kpeter@768: function. alpar@40: kpeter@768: LEMON contains three algorithms for solving the minimum mean cycle problem: alpar@1053: - \ref KarpMmc Karp's original algorithm \cite karp78characterization. kpeter@879: - \ref HartmannOrlinMmc Hartmann-Orlin's algorithm, which is an improved alpar@1053: version of Karp's algorithm \cite hartmann93finding. kpeter@879: - \ref HowardMmc Howard's policy iteration algorithm alpar@1053: \cite dasdan98minmeancycle, \cite dasdan04experimental. alpar@40: kpeter@919: In practice, the \ref HowardMmc "Howard" algorithm turned out to be by far the kpeter@879: most efficient one, though the best known theoretical bound on its running kpeter@879: time is exponential. kpeter@879: Both \ref KarpMmc "Karp" and \ref HartmannOrlinMmc "Hartmann-Orlin" algorithms kpeter@1080: run in time O(nm) and use space O(n2+m). 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@590: This group contains the algorithms for calculating alpar@40: matchings in graphs and bipartite graphs. The general matching problem is kpeter@590: finding a subset of the edges for which each node has at most one incident kpeter@590: 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. deba@869: - \ref MaxFractionalMatching Push-relabel algorithm for calculating deba@869: maximum cardinality fractional matching in general graphs. deba@869: - \ref MaxWeightedFractionalMatching Augmenting path algorithm for calculating deba@869: maximum weighted fractional matching in general graphs. deba@869: - \ref MaxWeightedPerfectFractionalMatching deba@869: Augmenting path algorithm for calculating maximum weighted deba@869: perfect fractional matching in general graphs. alpar@40: alpar@865: \image html matching.png alpar@873: \image latex matching.eps "Min Cost Perfect Matching" width=\textwidth alpar@40: */ alpar@40: alpar@40: /** kpeter@714: @defgroup graph_properties Connectivity and Other Graph Properties alpar@40: @ingroup algs kpeter@714: \brief Algorithms for discovering the graph properties alpar@40: kpeter@714: This group contains the algorithms for discovering the graph properties kpeter@714: like connectivity, bipartiteness, euler property, simplicity etc. kpeter@714: kpeter@714: \image html connected_components.png kpeter@714: \image latex connected_components.eps "Connected components" width=\textwidth kpeter@714: */ kpeter@714: kpeter@714: /** alpar@1142: @defgroup graph_isomorphism Graph Isomorphism alpar@1142: @ingroup algs alpar@1142: \brief Algorithms for testing (sub)graph isomorphism alpar@1142: alpar@1142: This group contains algorithms for finding isomorph copies of a alpar@1142: given graph in another one, or simply check whether two graphs are isomorphic. alpar@1142: alpar@1142: The formal definition of subgraph isomorphism is as follows. alpar@1142: alpar@1142: We are given two graphs, \f$G_1=(V_1,E_1)\f$ and \f$G_2=(V_2,E_2)\f$. A alpar@1142: function \f$f:V_1\longrightarrow V_2\f$ is called \e mapping or \e alpar@1142: embedding if \f$f(u)\neq f(v)\f$ whenever \f$u\neq v\f$. alpar@1142: alpar@1142: The standard Subgraph Isomorphism Problem (SIP) looks for a alpar@1142: mapping with the property that whenever \f$(u,v)\in E_1\f$, then alpar@1142: \f$(f(u),f(v))\in E_2\f$. alpar@1142: alpar@1142: In case of Induced Subgraph Isomorphism Problem (ISIP) one alpar@1142: also requires that if \f$(u,v)\not\in E_1\f$, then \f$(f(u),f(v))\not\in alpar@1142: E_2\f$ alpar@1142: alpar@1142: In addition, the graph nodes may be \e labeled, i.e. we are given two alpar@1142: node labelings \f$l_1:V_1\longrightarrow L\f$ and \f$l_2:V_2\longrightarrow alpar@1142: L\f$ and we require that \f$l_1(u)=l_2(f(u))\f$ holds for all nodes \f$u \in alpar@1142: G\f$. alpar@1142: alpar@1142: */ alpar@1142: alpar@1142: /** kpeter@919: @defgroup planar Planar Embedding and Drawing kpeter@714: @ingroup algs kpeter@714: \brief Algorithms for planarity checking, embedding and drawing kpeter@714: kpeter@714: This group contains the algorithms for planarity checking, kpeter@714: embedding and drawing. kpeter@714: kpeter@714: \image html planar.png kpeter@714: \image latex planar.eps "Plane graph" width=\textwidth kpeter@714: */ alpar@1092: kpeter@1032: /** kpeter@1032: @defgroup tsp Traveling Salesman Problem kpeter@1032: @ingroup algs kpeter@1032: \brief Algorithms for the symmetric traveling salesman problem kpeter@1032: kpeter@1032: This group contains basic heuristic algorithms for the the symmetric kpeter@1032: \e traveling \e salesman \e problem (TSP). kpeter@1032: Given an \ref FullGraph "undirected full graph" with a cost map on its edges, kpeter@1032: the problem is to find a shortest possible tour that visits each node exactly kpeter@1032: once (i.e. the minimum cost Hamiltonian cycle). kpeter@1032: kpeter@1034: These TSP algorithms are intended to be used with a \e metric \e cost kpeter@1034: \e function, i.e. the edge costs should satisfy the triangle inequality. kpeter@1034: Otherwise the algorithms could yield worse results. kpeter@1032: kpeter@1032: LEMON provides five well-known heuristics for solving symmetric TSP: kpeter@1032: - \ref NearestNeighborTsp Neareast neighbor algorithm kpeter@1032: - \ref GreedyTsp Greedy algorithm kpeter@1032: - \ref InsertionTsp Insertion heuristic (with four selection methods) kpeter@1032: - \ref ChristofidesTsp Christofides algorithm kpeter@1032: - \ref Opt2Tsp 2-opt algorithm kpeter@1032: kpeter@1036: \ref NearestNeighborTsp, \ref GreedyTsp, and \ref InsertionTsp are the fastest kpeter@1036: solution methods. Furthermore, \ref InsertionTsp is usually quite effective. kpeter@1036: kpeter@1036: \ref ChristofidesTsp is somewhat slower, but it has the best guaranteed kpeter@1036: approximation factor: 3/2. kpeter@1036: kpeter@1036: \ref Opt2Tsp usually provides the best results in practice, but kpeter@1036: it is the slowest method. It can also be used to improve given tours, kpeter@1036: for example, the results of other algorithms. kpeter@1036: kpeter@1032: \image html tsp.png kpeter@1032: \image latex tsp.eps "Traveling salesman problem" width=\textwidth kpeter@1032: */ kpeter@714: kpeter@714: /** kpeter@904: @defgroup approx_algs Approximation Algorithms kpeter@714: @ingroup algs kpeter@714: \brief Approximation algorithms. kpeter@714: kpeter@714: This group contains the approximation and heuristic algorithms kpeter@714: implemented in LEMON. kpeter@904: kpeter@904: Maximum Clique Problem kpeter@904: - \ref GrossoLocatelliPullanMc An efficient heuristic algorithm of kpeter@904: Grosso, Locatelli, and Pullan. 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@559: 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: /** alpar@40: @defgroup gen_opt_group General Optimization Tools kpeter@559: \brief This group contains some general optimization frameworks alpar@40: implemented in LEMON. alpar@40: kpeter@559: This group contains some general optimization frameworks alpar@40: implemented in LEMON. alpar@40: */ alpar@40: alpar@40: /** kpeter@755: @defgroup lp_group LP and MIP Solvers alpar@40: @ingroup gen_opt_group kpeter@755: \brief LP and MIP solver interfaces for LEMON. alpar@40: kpeter@755: This group contains LP and MIP solver interfaces for LEMON. kpeter@755: Various LP solvers could be used in the same manner with this kpeter@755: high-level interface. kpeter@755: alpar@1053: The currently supported solvers are \cite glpk, \cite clp, \cite cbc, alpar@1053: \cite cplex, \cite soplex. 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@559: 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@559: 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@559: 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@559: 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@559: 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@559: 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@559: 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@559: This group contains general \c EPS drawing methods and special alpar@209: graph exporting tools. kpeter@1050: kpeter@1050: \image html graph_to_eps.png alpar@40: */ alpar@40: alpar@40: /** kpeter@714: @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@559: 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, typedefs 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 checker class 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@735: This group contains the skeletons and concept checking classes of kpeter@735: graph structures. 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@559: This group contains the skeletons and concept checking classes of maps. alpar@40: */ alpar@40: alpar@40: /** kpeter@714: @defgroup tools Standalone Utility Applications kpeter@714: kpeter@714: Some utility applications are listed here. kpeter@714: kpeter@714: The standard compilation procedure (./configure;make) will compile kpeter@714: them, as well. kpeter@714: */ kpeter@714: kpeter@714: /** 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@564: In order to compile them, use the make demo or the ladanyi@564: make check commands. alpar@40: */ alpar@40: kpeter@406: }