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doc/groups.dox

author | Alpar Juttner <alpar@cs.elte.hu> |

Wed, 29 Jul 2020 14:56:10 +0200 | |

changeset 1433 | a278d16bd2d0 |

parent 1351 | 2f479109a71d |

permissions | -rw-r--r-- |

Fix clang compilation issue (#634)

1 /* -*- mode: C++; indent-tabs-mode: nil; -*-

2 *

3 * This file is a part of LEMON, a generic C++ optimization library.

4 *

5 * Copyright (C) 2003-2013

6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

7 * (Egervary Research Group on Combinatorial Optimization, EGRES).

8 *

9 * Permission to use, modify and distribute this software is granted

10 * provided that this copyright notice appears in all copies. For

11 * precise terms see the accompanying LICENSE file.

12 *

13 * This software is provided "AS IS" with no warranty of any kind,

14 * express or implied, and with no claim as to its suitability for any

15 * purpose.

16 *

17 */

19 namespace lemon {

21 /**

22 @defgroup datas Data Structures

23 This group contains the several data structures implemented in LEMON.

24 */

26 /**

27 @defgroup graphs Graph Structures

28 @ingroup datas

29 \brief Graph structures implemented in LEMON.

31 The implementation of combinatorial algorithms heavily relies on

32 efficient graph implementations. LEMON offers data structures which are

33 planned to be easily used in an experimental phase of implementation studies,

34 and thereafter the program code can be made efficient by small modifications.

36 The most efficient implementation of diverse applications require the

37 usage of different physical graph implementations. These differences

38 appear in the size of graph we require to handle, memory or time usage

39 limitations or in the set of operations through which the graph can be

40 accessed. LEMON provides several physical graph structures to meet

41 the diverging requirements of the possible users. In order to save on

42 running time or on memory usage, some structures may fail to provide

43 some graph features like arc/edge or node deletion.

45 Alteration of standard containers need a very limited number of

46 operations, these together satisfy the everyday requirements.

47 In the case of graph structures, different operations are needed which do

48 not alter the physical graph, but gives another view. If some nodes or

49 arcs have to be hidden or the reverse oriented graph have to be used, then

50 this is the case. It also may happen that in a flow implementation

51 the residual graph can be accessed by another algorithm, or a node-set

52 is to be shrunk for another algorithm.

53 LEMON also provides a variety of graphs for these requirements called

54 \ref graph_adaptors "graph adaptors". Adaptors cannot be used alone but only

55 in conjunction with other graph representations.

57 You are free to use the graph structure that fit your requirements

58 the best, most graph algorithms and auxiliary data structures can be used

59 with any graph structure.

61 <b>See also:</b> \ref graph_concepts "Graph Structure Concepts".

62 */

64 /**

65 @defgroup graph_adaptors Adaptor Classes for Graphs

66 @ingroup graphs

67 \brief Adaptor classes for digraphs and graphs

69 This group contains several useful adaptor classes for digraphs and graphs.

71 The main parts of LEMON are the different graph structures, generic

72 graph algorithms, graph concepts, which couple them, and graph

73 adaptors. While the previous notions are more or less clear, the

74 latter one needs further explanation. Graph adaptors are graph classes

75 which serve for considering graph structures in different ways.

77 A short example makes this much clearer. Suppose that we have an

78 instance \c g of a directed graph type, say ListDigraph and an algorithm

79 \code

80 template <typename Digraph>

81 int algorithm(const Digraph&);

82 \endcode

83 is needed to run on the reverse oriented graph. It may be expensive

84 (in time or in memory usage) to copy \c g with the reversed

85 arcs. In this case, an adaptor class is used, which (according

86 to LEMON \ref concepts::Digraph "digraph concepts") works as a digraph.

87 The adaptor uses the original digraph structure and digraph operations when

88 methods of the reversed oriented graph are called. This means that the adaptor

89 have minor memory usage, and do not perform sophisticated algorithmic

90 actions. The purpose of it is to give a tool for the cases when a

91 graph have to be used in a specific alteration. If this alteration is

92 obtained by a usual construction like filtering the node or the arc set or

93 considering a new orientation, then an adaptor is worthwhile to use.

94 To come back to the reverse oriented graph, in this situation

95 \code

96 template<typename Digraph> class ReverseDigraph;

97 \endcode

98 template class can be used. The code looks as follows

99 \code

100 ListDigraph g;

101 ReverseDigraph<ListDigraph> rg(g);

102 int result = algorithm(rg);

103 \endcode

104 During running the algorithm, the original digraph \c g is untouched.

105 This techniques give rise to an elegant code, and based on stable

106 graph adaptors, complex algorithms can be implemented easily.

108 In flow, circulation and matching problems, the residual

109 graph is of particular importance. Combining an adaptor implementing

110 this with shortest path algorithms or minimum mean cycle algorithms,

111 a range of weighted and cardinality optimization algorithms can be

112 obtained. For other examples, the interested user is referred to the

113 detailed documentation of particular adaptors.

115 Since the adaptor classes conform to the \ref graph_concepts "graph concepts",

116 an adaptor can even be applied to another one.

117 The following image illustrates a situation when a \ref SubDigraph adaptor

118 is applied on a digraph and \ref Undirector is applied on the subgraph.

120 \image html adaptors2.png

121 \image latex adaptors2.eps "Using graph adaptors" width=\textwidth

123 The behavior of graph adaptors can be very different. Some of them keep

124 capabilities of the original graph while in other cases this would be

125 meaningless. This means that the concepts that they meet depend

126 on the graph adaptor, and the wrapped graph.

127 For example, if an arc of a reversed digraph is deleted, this is carried

128 out by deleting the corresponding arc of the original digraph, thus the

129 adaptor modifies the original digraph.

130 However in case of a residual digraph, this operation has no sense.

132 Let us stand one more example here to simplify your work.

133 ReverseDigraph has constructor

134 \code

135 ReverseDigraph(Digraph& digraph);

136 \endcode

137 This means that in a situation, when a <tt>const %ListDigraph&</tt>

138 reference to a graph is given, then it have to be instantiated with

139 <tt>Digraph=const %ListDigraph</tt>.

140 \code

141 int algorithm1(const ListDigraph& g) {

142 ReverseDigraph<const ListDigraph> rg(g);

143 return algorithm2(rg);

144 }

145 \endcode

146 */

148 /**

149 @defgroup maps Maps

150 @ingroup datas

151 \brief Map structures implemented in LEMON.

153 This group contains the map structures implemented in LEMON.

155 LEMON provides several special purpose maps and map adaptors that e.g. combine

156 new maps from existing ones.

158 <b>See also:</b> \ref map_concepts "Map Concepts".

159 */

161 /**

162 @defgroup graph_maps Graph Maps

163 @ingroup maps

164 \brief Special graph-related maps.

166 This group contains maps that are specifically designed to assign

167 values to the nodes and arcs/edges of graphs.

169 If you are looking for the standard graph maps (\c NodeMap, \c ArcMap,

170 \c EdgeMap), see the \ref graph_concepts "Graph Structure Concepts".

171 */

173 /**

174 \defgroup map_adaptors Map Adaptors

175 \ingroup maps

176 \brief Tools to create new maps from existing ones

178 This group contains map adaptors that are used to create "implicit"

179 maps from other maps.

181 Most of them are \ref concepts::ReadMap "read-only maps".

182 They can make arithmetic and logical operations between one or two maps

183 (negation, shifting, addition, multiplication, logical 'and', 'or',

184 'not' etc.) or e.g. convert a map to another one of different Value type.

186 The typical usage of this classes is passing implicit maps to

187 algorithms. If a function type algorithm is called then the function

188 type map adaptors can be used comfortable. For example let's see the

189 usage of map adaptors with the \c graphToEps() function.

190 \code

191 Color nodeColor(int deg) {

192 if (deg >= 2) {

193 return Color(0.5, 0.0, 0.5);

194 } else if (deg == 1) {

195 return Color(1.0, 0.5, 1.0);

196 } else {

197 return Color(0.0, 0.0, 0.0);

198 }

199 }

201 Digraph::NodeMap<int> degree_map(graph);

203 graphToEps(graph, "graph.eps")

204 .coords(coords).scaleToA4().undirected()

205 .nodeColors(composeMap(functorToMap(nodeColor), degree_map))

206 .run();

207 \endcode

208 The \c functorToMap() function makes an \c int to \c Color map from the

209 \c nodeColor() function. The \c composeMap() compose the \c degree_map

210 and the previously created map. The composed map is a proper function to

211 get the color of each node.

213 The usage with class type algorithms is little bit harder. In this

214 case the function type map adaptors can not be used, because the

215 function map adaptors give back temporary objects.

216 \code

217 Digraph graph;

219 typedef Digraph::ArcMap<double> DoubleArcMap;

220 DoubleArcMap length(graph);

221 DoubleArcMap speed(graph);

223 typedef DivMap<DoubleArcMap, DoubleArcMap> TimeMap;

224 TimeMap time(length, speed);

226 Dijkstra<Digraph, TimeMap> dijkstra(graph, time);

227 dijkstra.run(source, target);

228 \endcode

229 We have a length map and a maximum speed map on the arcs of a digraph.

230 The minimum time to pass the arc can be calculated as the division of

231 the two maps which can be done implicitly with the \c DivMap template

232 class. We use the implicit minimum time map as the length map of the

233 \c Dijkstra algorithm.

234 */

236 /**

237 @defgroup paths Path Structures

238 @ingroup datas

239 \brief %Path structures implemented in LEMON.

241 This group contains the path structures implemented in LEMON.

243 LEMON provides flexible data structures to work with paths.

244 All of them have similar interfaces and they can be copied easily with

245 assignment operators and copy constructors. This makes it easy and

246 efficient to have e.g. the Dijkstra algorithm to store its result in

247 any kind of path structure.

249 \sa \ref concepts::Path "Path concept"

250 */

252 /**

253 @defgroup heaps Heap Structures

254 @ingroup datas

255 \brief %Heap structures implemented in LEMON.

257 This group contains the heap structures implemented in LEMON.

259 LEMON provides several heap classes. They are efficient implementations

260 of the abstract data type \e priority \e queue. They store items with

261 specified values called \e priorities in such a way that finding and

262 removing the item with minimum priority are efficient.

263 The basic operations are adding and erasing items, changing the priority

264 of an item, etc.

266 Heaps are crucial in several algorithms, such as Dijkstra and Prim.

267 The heap implementations have the same interface, thus any of them can be

268 used easily in such algorithms.

270 \sa \ref concepts::Heap "Heap concept"

271 */

273 /**

274 @defgroup auxdat Auxiliary Data Structures

275 @ingroup datas

276 \brief Auxiliary data structures implemented in LEMON.

278 This group contains some data structures implemented in LEMON in

279 order to make it easier to implement combinatorial algorithms.

280 */

282 /**

283 @defgroup geomdat Geometric Data Structures

284 @ingroup auxdat

285 \brief Geometric data structures implemented in LEMON.

287 This group contains geometric data structures implemented in LEMON.

289 - \ref lemon::dim2::Point "dim2::Point" implements a two dimensional

290 vector with the usual operations.

291 - \ref lemon::dim2::Box "dim2::Box" can be used to determine the

292 rectangular bounding box of a set of \ref lemon::dim2::Point

293 "dim2::Point"'s.

294 */

296 /**

297 @defgroup matrices Matrices

298 @ingroup auxdat

299 \brief Two dimensional data storages implemented in LEMON.

301 This group contains two dimensional data storages implemented in LEMON.

302 */

304 /**

305 @defgroup algs Algorithms

306 \brief This group contains the several algorithms

307 implemented in LEMON.

309 This group contains the several algorithms

310 implemented in LEMON.

311 */

313 /**

314 @defgroup search Graph Search

315 @ingroup algs

316 \brief Common graph search algorithms.

318 This group contains the common graph search algorithms, namely

319 \e breadth-first \e search (BFS) and \e depth-first \e search (DFS)

320 \cite clrs01algorithms.

321 */

323 /**

324 @defgroup shortest_path Shortest Path Algorithms

325 @ingroup algs

326 \brief Algorithms for finding shortest paths.

328 This group contains the algorithms for finding shortest paths in digraphs

329 \cite clrs01algorithms.

331 - \ref Dijkstra algorithm for finding shortest paths from a source node

332 when all arc lengths are non-negative.

333 - \ref BellmanFord "Bellman-Ford" algorithm for finding shortest paths

334 from a source node when arc lenghts can be either positive or negative,

335 but the digraph should not contain directed cycles with negative total

336 length.

337 - \ref FloydWarshall "Floyd-Warshall" and \ref Johnson "Johnson" algorithms

338 for solving the \e all-pairs \e shortest \e paths \e problem when arc

339 lenghts can be either positive or negative, but the digraph should

340 not contain directed cycles with negative total length.

341 - \ref Suurballe A successive shortest path algorithm for finding

342 arc-disjoint paths between two nodes having minimum total length.

343 */

345 /**

346 @defgroup spantree Minimum Spanning Tree Algorithms

347 @ingroup algs

348 \brief Algorithms for finding minimum cost spanning trees and arborescences.

350 This group contains the algorithms for finding minimum cost spanning

351 trees and arborescences \cite clrs01algorithms.

352 */

354 /**

355 @defgroup max_flow Maximum Flow Algorithms

356 @ingroup algs

357 \brief Algorithms for finding maximum flows.

359 This group contains the algorithms for finding maximum flows and

360 feasible circulations \cite clrs01algorithms, \cite amo93networkflows.

362 The \e maximum \e flow \e problem is to find a flow of maximum value between

363 a single source and a single target. Formally, there is a \f$G=(V,A)\f$

364 digraph, a \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function and

365 \f$s, t \in V\f$ source and target nodes.

366 A maximum flow is an \f$f: A\rightarrow\mathbf{R}^+_0\f$ solution of the

367 following optimization problem.

369 \f[ \max\sum_{sv\in A} f(sv) - \sum_{vs\in A} f(vs) \f]

370 \f[ \sum_{uv\in A} f(uv) = \sum_{vu\in A} f(vu)

371 \quad \forall u\in V\setminus\{s,t\} \f]

372 \f[ 0 \leq f(uv) \leq cap(uv) \quad \forall uv\in A \f]

374 LEMON contains several algorithms for solving maximum flow problems:

375 - \ref EdmondsKarp Edmonds-Karp algorithm

376 \cite edmondskarp72theoretical.

377 - \ref Preflow Goldberg-Tarjan's preflow push-relabel algorithm

378 \cite goldberg88newapproach.

379 - \ref DinitzSleatorTarjan Dinitz's blocking flow algorithm with dynamic trees

380 \cite dinic70algorithm, \cite sleator83dynamic.

381 - \ref GoldbergTarjan !Preflow push-relabel algorithm with dynamic trees

382 \cite goldberg88newapproach, \cite sleator83dynamic.

384 In most cases the \ref Preflow algorithm provides the

385 fastest method for computing a maximum flow. All implementations

386 also provide functions to query the minimum cut, which is the dual

387 problem of maximum flow.

389 \ref Circulation is a preflow push-relabel algorithm implemented directly

390 for finding feasible circulations, which is a somewhat different problem,

391 but it is strongly related to maximum flow.

392 For more information, see \ref Circulation.

393 */

395 /**

396 @defgroup min_cost_flow_algs Minimum Cost Flow Algorithms

397 @ingroup algs

399 \brief Algorithms for finding minimum cost flows and circulations.

401 This group contains the algorithms for finding minimum cost flows and

402 circulations \cite amo93networkflows. For more information about this

403 problem and its dual solution, see: \ref min_cost_flow

404 "Minimum Cost Flow Problem".

406 LEMON contains several algorithms for this problem.

407 - \ref NetworkSimplex Primal Network Simplex algorithm with various

408 pivot strategies \cite dantzig63linearprog, \cite kellyoneill91netsimplex.

409 - \ref CostScaling Cost Scaling algorithm based on push/augment and

410 relabel operations \cite goldberg90approximation, \cite goldberg97efficient,

411 \cite bunnagel98efficient.

412 - \ref CapacityScaling Capacity Scaling algorithm based on the successive

413 shortest path method \cite edmondskarp72theoretical.

414 - \ref CycleCanceling Cycle-Canceling algorithms, two of which are

415 strongly polynomial \cite klein67primal, \cite goldberg89cyclecanceling.

417 In general, \ref NetworkSimplex and \ref CostScaling are the most efficient

418 implementations.

419 \ref NetworkSimplex is usually the fastest on relatively small graphs (up to

420 several thousands of nodes) and on dense graphs, while \ref CostScaling is

421 typically more efficient on large graphs (e.g. hundreds of thousands of

422 nodes or above), especially if they are sparse.

423 However, other algorithms could be faster in special cases.

424 For example, if the total supply and/or capacities are rather small,

425 \ref CapacityScaling is usually the fastest algorithm

426 (without effective scaling).

428 These classes are intended to be used with integer-valued input data

429 (capacities, supply values, and costs), except for \ref CapacityScaling,

430 which is capable of handling real-valued arc costs (other numerical

431 data are required to be integer).

433 For more details about these implementations and for a comprehensive

434 experimental study, see the paper \cite KiralyKovacs12MCF.

435 It also compares these codes to other publicly available

436 minimum cost flow solvers.

437 */

439 /**

440 @defgroup min_cut Minimum Cut Algorithms

441 @ingroup algs

443 \brief Algorithms for finding minimum cut in graphs.

445 This group contains the algorithms for finding minimum cut in graphs.

447 The \e minimum \e cut \e problem is to find a non-empty and non-complete

448 \f$X\f$ subset of the nodes with minimum overall capacity on

449 outgoing arcs. Formally, there is a \f$G=(V,A)\f$ digraph, a

450 \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function. The minimum

451 cut is the \f$X\f$ solution of the next optimization problem:

453 \f[ \min_{X \subset V, X\not\in \{\emptyset, V\}}

454 \sum_{uv\in A: u\in X, v\not\in X}cap(uv) \f]

456 LEMON contains several algorithms related to minimum cut problems:

458 - \ref HaoOrlin "Hao-Orlin algorithm" for calculating minimum cut

459 in directed graphs.

460 - \ref NagamochiIbaraki "Nagamochi-Ibaraki algorithm" for

461 calculating minimum cut in undirected graphs.

462 - \ref GomoryHu "Gomory-Hu tree computation" for calculating

463 all-pairs minimum cut in undirected graphs.

465 If you want to find minimum cut just between two distinict nodes,

466 see the \ref max_flow "maximum flow problem".

467 */

469 /**

470 @defgroup min_mean_cycle Minimum Mean Cycle Algorithms

471 @ingroup algs

472 \brief Algorithms for finding minimum mean cycles.

474 This group contains the algorithms for finding minimum mean cycles

475 \cite amo93networkflows, \cite karp78characterization.

477 The \e minimum \e mean \e cycle \e problem is to find a directed cycle

478 of minimum mean length (cost) in a digraph.

479 The mean length of a cycle is the average length of its arcs, i.e. the

480 ratio between the total length of the cycle and the number of arcs on it.

482 This problem has an important connection to \e conservative \e length

483 \e functions, too. A length function on the arcs of a digraph is called

484 conservative if and only if there is no directed cycle of negative total

485 length. For an arbitrary length function, the negative of the minimum

486 cycle mean is the smallest \f$\epsilon\f$ value so that increasing the

487 arc lengths uniformly by \f$\epsilon\f$ results in a conservative length

488 function.

490 LEMON contains three algorithms for solving the minimum mean cycle problem:

491 - \ref KarpMmc Karp's original algorithm \cite karp78characterization.

492 - \ref HartmannOrlinMmc Hartmann-Orlin's algorithm, which is an improved

493 version of Karp's algorithm \cite hartmann93finding.

494 - \ref HowardMmc Howard's policy iteration algorithm

495 \cite dasdan98minmeancycle, \cite dasdan04experimental.

497 In practice, the \ref HowardMmc "Howard" algorithm turned out to be by far the

498 most efficient one, though the best known theoretical bound on its running

499 time is exponential.

500 Both \ref KarpMmc "Karp" and \ref HartmannOrlinMmc "Hartmann-Orlin" algorithms

501 run in time O(nm) and use space O(n<sup>2</sup>+m).

502 */

504 /**

505 @defgroup matching Matching Algorithms

506 @ingroup algs

507 \brief Algorithms for finding matchings in graphs and bipartite graphs.

509 This group contains the algorithms for calculating

510 matchings in graphs and bipartite graphs. The general matching problem is

511 finding a subset of the edges for which each node has at most one incident

512 edge.

514 There are several different algorithms for calculate matchings in

515 graphs. The matching problems in bipartite graphs are generally

516 easier than in general graphs. The goal of the matching optimization

517 can be finding maximum cardinality, maximum weight or minimum cost

518 matching. The search can be constrained to find perfect or

519 maximum cardinality matching.

521 The matching algorithms implemented in LEMON:

522 - \ref MaxBipartiteMatching Hopcroft-Karp augmenting path algorithm

523 for calculating maximum cardinality matching in bipartite graphs.

524 - \ref PrBipartiteMatching Push-relabel algorithm

525 for calculating maximum cardinality matching in bipartite graphs.

526 - \ref MaxWeightedBipartiteMatching

527 Successive shortest path algorithm for calculating maximum weighted

528 matching and maximum weighted bipartite matching in bipartite graphs.

529 - \ref MinCostMaxBipartiteMatching

530 Successive shortest path algorithm for calculating minimum cost maximum

531 matching in bipartite graphs.

532 - \ref MaxMatching Edmond's blossom shrinking algorithm for calculating

533 maximum cardinality matching in general graphs.

534 - \ref MaxWeightedMatching Edmond's blossom shrinking algorithm for calculating

535 maximum weighted matching in general graphs.

536 - \ref MaxWeightedPerfectMatching

537 Edmond's blossom shrinking algorithm for calculating maximum weighted

538 perfect matching in general graphs.

539 - \ref MaxFractionalMatching Push-relabel algorithm for calculating

540 maximum cardinality fractional matching in general graphs.

541 - \ref MaxWeightedFractionalMatching Augmenting path algorithm for calculating

542 maximum weighted fractional matching in general graphs.

543 - \ref MaxWeightedPerfectFractionalMatching

544 Augmenting path algorithm for calculating maximum weighted

545 perfect fractional matching in general graphs.

547 \image html matching.png

548 \image latex matching.eps "Min Cost Perfect Matching" width=\textwidth

549 */

551 /**

552 @defgroup graph_properties Connectivity and Other Graph Properties

553 @ingroup algs

554 \brief Algorithms for discovering the graph properties

556 This group contains the algorithms for discovering the graph properties

557 like connectivity, bipartiteness, euler property, simplicity etc.

559 \image html connected_components.png

560 \image latex connected_components.eps "Connected components" width=\textwidth

561 */

563 /**

564 @defgroup graph_isomorphism Graph Isomorphism

565 @ingroup algs

566 \brief Algorithms for testing (sub)graph isomorphism

568 This group contains algorithms for finding isomorph copies of a

569 given graph in another one, or simply check whether two graphs are isomorphic.

571 The formal definition of subgraph isomorphism is as follows.

573 We are given two graphs, \f$G_1=(V_1,E_1)\f$ and \f$G_2=(V_2,E_2)\f$. A

574 function \f$f:V_1\longrightarrow V_2\f$ is called \e mapping or \e

575 embedding if \f$f(u)\neq f(v)\f$ whenever \f$u\neq v\f$.

577 The standard <em>Subgraph Isomorphism Problem (SIP)</em> looks for a

578 mapping with the property that whenever \f$(u,v)\in E_1\f$, then

579 \f$(f(u),f(v))\in E_2\f$.

581 In case of <em>Induced Subgraph Isomorphism Problem (ISIP)</em> one

582 also requires that if \f$(u,v)\not\in E_1\f$, then \f$(f(u),f(v))\not\in

583 E_2\f$

585 In addition, the graph nodes may be \e labeled, i.e. we are given two

586 node labelings \f$l_1:V_1\longrightarrow L\f$ and \f$l_2:V_2\longrightarrow

587 L\f$ and we require that \f$l_1(u)=l_2(f(u))\f$ holds for all nodes \f$u \in

588 G_1\f$.

590 */

592 /**

593 @defgroup planar Planar Embedding and Drawing

594 @ingroup algs

595 \brief Algorithms for planarity checking, embedding and drawing

597 This group contains the algorithms for planarity checking,

598 embedding and drawing.

600 \image html planar.png

601 \image latex planar.eps "Plane graph" width=\textwidth

602 */

604 /**

605 @defgroup tsp Traveling Salesman Problem

606 @ingroup algs

607 \brief Algorithms for the symmetric traveling salesman problem

609 This group contains basic heuristic algorithms for the the symmetric

610 \e traveling \e salesman \e problem (TSP).

611 Given an \ref FullGraph "undirected full graph" with a cost map on its edges,

612 the problem is to find a shortest possible tour that visits each node exactly

613 once (i.e. the minimum cost Hamiltonian cycle).

615 These TSP algorithms are intended to be used with a \e metric \e cost

616 \e function, i.e. the edge costs should satisfy the triangle inequality.

617 Otherwise the algorithms could yield worse results.

619 LEMON provides five well-known heuristics for solving symmetric TSP:

620 - \ref NearestNeighborTsp Neareast neighbor algorithm

621 - \ref GreedyTsp Greedy algorithm

622 - \ref InsertionTsp Insertion heuristic (with four selection methods)

623 - \ref ChristofidesTsp Christofides algorithm

624 - \ref Opt2Tsp 2-opt algorithm

626 \ref NearestNeighborTsp, \ref GreedyTsp, and \ref InsertionTsp are the fastest

627 solution methods. Furthermore, \ref InsertionTsp is usually quite effective.

629 \ref ChristofidesTsp is somewhat slower, but it has the best guaranteed

630 approximation factor: 3/2.

632 \ref Opt2Tsp usually provides the best results in practice, but

633 it is the slowest method. It can also be used to improve given tours,

634 for example, the results of other algorithms.

636 \image html tsp.png

637 \image latex tsp.eps "Traveling salesman problem" width=\textwidth

638 */

640 /**

641 @defgroup approx_algs Approximation Algorithms

642 @ingroup algs

643 \brief Approximation algorithms.

645 This group contains the approximation and heuristic algorithms

646 implemented in LEMON.

648 <b>Maximum Clique Problem</b>

649 - \ref GrossoLocatelliPullanMc An efficient heuristic algorithm of

650 Grosso, Locatelli, and Pullan.

651 */

653 /**

654 @defgroup auxalg Auxiliary Algorithms

655 @ingroup algs

656 \brief Auxiliary algorithms implemented in LEMON.

658 This group contains some algorithms implemented in LEMON

659 in order to make it easier to implement complex algorithms.

660 */

662 /**

663 @defgroup gen_opt_group General Optimization Tools

664 \brief This group contains some general optimization frameworks

665 implemented in LEMON.

667 This group contains some general optimization frameworks

668 implemented in LEMON.

669 */

671 /**

672 @defgroup lp_group LP and MIP Solvers

673 @ingroup gen_opt_group

674 \brief LP and MIP solver interfaces for LEMON.

676 This group contains LP and MIP solver interfaces for LEMON.

677 Various LP solvers could be used in the same manner with this

678 high-level interface.

680 The currently supported solvers are \cite glpk, \cite clp, \cite cbc,

681 \cite cplex, \cite soplex.

682 */

684 /**

685 @defgroup lp_utils Tools for Lp and Mip Solvers

686 @ingroup lp_group

687 \brief Helper tools to the Lp and Mip solvers.

689 This group adds some helper tools to general optimization framework

690 implemented in LEMON.

691 */

693 /**

694 @defgroup metah Metaheuristics

695 @ingroup gen_opt_group

696 \brief Metaheuristics for LEMON library.

698 This group contains some metaheuristic optimization tools.

699 */

701 /**

702 @defgroup utils Tools and Utilities

703 \brief Tools and utilities for programming in LEMON

705 Tools and utilities for programming in LEMON.

706 */

708 /**

709 @defgroup gutils Basic Graph Utilities

710 @ingroup utils

711 \brief Simple basic graph utilities.

713 This group contains some simple basic graph utilities.

714 */

716 /**

717 @defgroup misc Miscellaneous Tools

718 @ingroup utils

719 \brief Tools for development, debugging and testing.

721 This group contains several useful tools for development,

722 debugging and testing.

723 */

725 /**

726 @defgroup timecount Time Measuring and Counting

727 @ingroup misc

728 \brief Simple tools for measuring the performance of algorithms.

730 This group contains simple tools for measuring the performance

731 of algorithms.

732 */

734 /**

735 @defgroup exceptions Exceptions

736 @ingroup utils

737 \brief Exceptions defined in LEMON.

739 This group contains the exceptions defined in LEMON.

740 */

742 /**

743 @defgroup io_group Input-Output

744 \brief Graph Input-Output methods

746 This group contains the tools for importing and exporting graphs

747 and graph related data. Now it supports the \ref lgf-format

748 "LEMON Graph Format", the \c DIMACS format and the encapsulated

749 postscript (EPS) format.

750 */

752 /**

753 @defgroup lemon_io LEMON Graph Format

754 @ingroup io_group

755 \brief Reading and writing LEMON Graph Format.

757 This group contains methods for reading and writing

758 \ref lgf-format "LEMON Graph Format".

759 */

761 /**

762 @defgroup eps_io Postscript Exporting

763 @ingroup io_group

764 \brief General \c EPS drawer and graph exporter

766 This group contains general \c EPS drawing methods and special

767 graph exporting tools.

769 \image html graph_to_eps.png

770 */

772 /**

773 @defgroup dimacs_group DIMACS Format

774 @ingroup io_group

775 \brief Read and write files in DIMACS format

777 Tools to read a digraph from or write it to a file in DIMACS format data.

778 */

780 /**

781 @defgroup nauty_group NAUTY Format

782 @ingroup io_group

783 \brief Read \e Nauty format

785 Tool to read graphs from \e Nauty format data.

786 */

788 /**

789 @defgroup concept Concepts

790 \brief Skeleton classes and concept checking classes

792 This group contains the data/algorithm skeletons and concept checking

793 classes implemented in LEMON.

795 The purpose of the classes in this group is fourfold.

797 - These classes contain the documentations of the %concepts. In order

798 to avoid document multiplications, an implementation of a concept

799 simply refers to the corresponding concept class.

801 - These classes declare every functions, <tt>typedef</tt>s etc. an

802 implementation of the %concepts should provide, however completely

803 without implementations and real data structures behind the

804 interface. On the other hand they should provide nothing else. All

805 the algorithms working on a data structure meeting a certain concept

806 should compile with these classes. (Though it will not run properly,

807 of course.) In this way it is easily to check if an algorithm

808 doesn't use any extra feature of a certain implementation.

810 - The concept descriptor classes also provide a <em>checker class</em>

811 that makes it possible to check whether a certain implementation of a

812 concept indeed provides all the required features.

814 - Finally, They can serve as a skeleton of a new implementation of a concept.

815 */

817 /**

818 @defgroup graph_concepts Graph Structure Concepts

819 @ingroup concept

820 \brief Skeleton and concept checking classes for graph structures

822 This group contains the skeletons and concept checking classes of

823 graph structures.

824 */

826 /**

827 @defgroup map_concepts Map Concepts

828 @ingroup concept

829 \brief Skeleton and concept checking classes for maps

831 This group contains the skeletons and concept checking classes of maps.

832 */

834 /**

835 @defgroup tools Standalone Utility Applications

837 Some utility applications are listed here.

839 The standard compilation procedure (<tt>./configure;make</tt>) will compile

840 them, as well.

841 */

843 /**

844 \anchor demoprograms

846 @defgroup demos Demo Programs

848 Some demo programs are listed here. Their full source codes can be found in

849 the \c demo subdirectory of the source tree.

851 In order to compile them, use the <tt>make demo</tt> or the

852 <tt>make check</tt> commands.

853 */

855 }