@ingroup algs

 @ingroup algs

 \brief Common graph search algorithms.

 \brief Common graph search algorithms.

 This group contains the common graph search algorithms, namely

 This group contains the common graph search algorithms, namely

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

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

 */

 */

 /**

 /**

 @defgroup shortest_path Shortest Path Algorithms

 @defgroup shortest_path Shortest Path Algorithms

 @ingroup algs

 @ingroup algs

 \brief Algorithms for finding shortest paths.

 \brief Algorithms for finding shortest paths.

 This group contains the algorithms for finding shortest paths in digraphs

 This group contains the algorithms for finding shortest paths in digraphs

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

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

    when all arc lengths are non-negative.

    when all arc lengths are non-negative.

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

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

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

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

 @defgroup spantree Minimum Spanning Tree Algorithms

 @defgroup spantree Minimum Spanning Tree Algorithms

 @ingroup algs

 @ingroup algs

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

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

 This group contains the algorithms for finding minimum cost spanning

 This group contains the algorithms for finding minimum cost spanning

 */

 */

 /**

 /**

 @defgroup max_flow Maximum Flow Algorithms

 @defgroup max_flow Maximum Flow Algorithms

 @ingroup algs

 @ingroup algs

 \brief Algorithms for finding maximum flows.

 \brief Algorithms for finding maximum flows.

 This group contains the algorithms for finding maximum flows and

 This group contains the algorithms for finding maximum flows and

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

 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$G=(V,A)\f$

 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

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

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

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

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

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

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

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

 LEMON contains several algorithms for solving maximum flow problems:

 LEMON contains several algorithms for solving maximum flow problems:

 - \ref EdmondsKarp Edmonds-Karp algorithm

 - \ref EdmondsKarp Edmonds-Karp algorithm

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

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

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

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

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

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

 In most cases the \ref Preflow algorithm provides the

 In most cases the \ref Preflow algorithm provides the

 fastest method for computing a maximum flow. All implementations

 fastest method for computing a maximum flow. All implementations

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

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

 problem of maximum flow.

 problem of maximum flow.

 @ingroup algs

 @ingroup algs

 \brief Algorithms for finding minimum cost flows and circulations.

 \brief Algorithms for finding minimum cost flows and circulations.

 This group contains the algorithms for finding minimum cost flows and

 This group contains the algorithms for finding minimum cost flows and

 problem and its dual solution, see: \ref min_cost_flow

 problem and its dual solution, see: \ref min_cost_flow

 "Minimum Cost Flow Problem".

 "Minimum Cost Flow Problem".

 LEMON contains several algorithms for this problem.

 LEMON contains several algorithms for this problem.

  - \ref NetworkSimplex Primal Network Simplex algorithm with various

  - \ref NetworkSimplex Primal Network Simplex algorithm with various

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

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

  - \ref CapacityScaling Capacity Scaling algorithm based on the successive

  - \ref CapacityScaling Capacity Scaling algorithm based on the successive

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

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

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

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

 implementations.

 implementations.

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

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

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

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

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

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

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

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

 data are required to be integer).

 data are required to be integer).

 For more details about these implementations and for a comprehensive 

 For more details about these implementations and for a comprehensive 

 It also compares these codes to other publicly available

 It also compares these codes to other publicly available

 minimum cost flow solvers.

 minimum cost flow solvers.

 */

 */

 /**

 /**

 @defgroup min_mean_cycle Minimum Mean Cycle Algorithms

 @defgroup min_mean_cycle Minimum Mean Cycle Algorithms

 @ingroup algs

 @ingroup algs

 \brief Algorithms for finding minimum mean cycles.

 \brief Algorithms for finding minimum mean cycles.

 This group contains the algorithms for finding minimum mean cycles

 This group contains the algorithms for finding minimum mean cycles

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

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

 of minimum mean length (cost) in a digraph.

 of minimum mean length (cost) in a digraph.

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

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

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

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

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

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

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

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

 function.

 function.

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

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

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

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

 - \ref HowardMmc Howard's policy iteration algorithm

 - \ref HowardMmc Howard's policy iteration algorithm

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

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

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

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

 time is exponential.

 time is exponential.

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

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

 This group contains LP and MIP solver interfaces for LEMON.

 This group contains LP and MIP solver interfaces for LEMON.

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

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

 high-level interface.

 high-level interface.

 */

 */

 /**

 /**

 @defgroup lp_utils Tools for Lp and Mip Solvers

 @defgroup lp_utils Tools for Lp and Mip Solvers

 @ingroup lp_group

 @ingroup lp_group