# `[maximum|minimum]_cycle_ratio`

    template <typename Graph, typename VertexIndexMap,
              typename EdgeWeight1, typename EdgeWeight2>
    dobule
    maximum_cycle_ratio(const Graph &g, VertexIndexMap vim,
                        EdgeWeight1 ewm, EdgeWeight2 ew2m,
                        std::vector<typename boost::graph_traits<Graph>::edge_descriptor> *pcc = 0);

    template <typename FloatTraits, Graph, typename VertexIndexMap,
              typename EdgeWeight1, typename EdgeWeight2>
    typename FloatTraits::float_type
    maximum_cycle_ratio(const Graph &g, VertexIndexMap vim,
                        EdgeWeight1 ewm, EdgeWeight2 ew2m,
                        std::vector<typename boost::graph_traits<Graph>::edge_descriptor> *pcc = 0,
                        FloatTraits = FloatTraits());

    template <typename Graph, typename VertexIndexMap,
              typename EdgeWeight1, typename EdgeWeight2>
    dobule
    minimum_cycle_ratio(const Graph &g, VertexIndexMap vim,
                        EdgeWeight1 ewm, EdgeWeight2 ew2m,
                        std::vector<typename boost::graph_traits<Graph>::edge_descriptor> *pcc = 0);

    template <typename FloatTraits, typename Graph, typename VertexIndexMap,
              typename EdgeWeight1, typename EdgeWeight2>
    typename FloatTraits::float_type
    minimum_cycle_ratio(const Graph &g, VertexIndexMap vim,
                        EdgeWeight1 ewm, EdgeWeight2 ew2m,
                        std::vector<typename boost::graph_traits<Graph>::edge_descriptor> *pcc = 0,
                        FloatTraits = FloatTraits());

The `maximum_cycle_ratio()` function calculates the maximum cycle ratio of a weighted directed multigraph G=(V,E,W1,W2), where V is a vertex set, E is an edge set, W1 and W2 are edge weight functions, W2 is nonnegative. As a multigraph, G can have multiple edges connecting a pair of vertices.

Let every edge e has two weights W1(e) and W2(e). Let c be a cycle of the graphg. Then, the cycle ratio, cr(c), is defined as:

The maximum (minimum) cycle ratio (mcr) is the maximum (minimum) cycle ratio of all cycles of the graph. A cycle is called critical if its ratio is equal to the mcr. The calculated maximum cycle ratio will be the return value of the function. The `maximum_cycle_ratio()/minimum_cycle_ratio()` returns `-FloatTraits::infinity()/FloatTraits::infinity()` if graph has no cycles. If the pcc parameter is not zero then one critical cycle will be written to the corresponding `std::vector` of edge descriptors. The edges in the vector stored in the way such that `*pcc[0], *ppc[1], ..., *ppc[n]` is a correct path.

The algorithm is due to Howard's iteration policy algorithm, descibed in [1]. Ali Dasdan, Sandy S. Irani and Rajesh K.Gupta in their paper Efficient Algorithms for Optimum Cycle Mean and Optimum Cost to Time Ratio Problemsstate that this is the most efficient algorithm to the time being (1999).

For the convenience, functions `maximum_cycle_mean()` and `minimum_cycle_mean()` are also provided. They have the following signatures:

    template <typename Graph, typename VertexIndexMap,
              typename EdgeWeightMap, typename EdgeIndexMap>
    double
    maximum_cycle_mean(const Graph &g, VertexIndexMap vim,
                       EdgeWeightMap ewm, EdgeIndexMap eim,
                       std::vector<typename graph_traits<Graph>::edge_descriptor> *pcc = 0);
    template <typename FloatTraits, typename Graph, typename VertexIndexMap,
              typename EdgeWeightMap, typename EdgeIndexMap>

    typename FloatTraits::float_type
    maximum_cycle_mean(const Graph &g, VertexIndexMap vim,
                       EdgeWeightMap ewm, EdgeIndexMap eim,
                       std::vector<typename graph_traits<Graph>::edge_descriptor> *pcc = 0,
                       FloatTraits = FloatTraits());

    template <typename Graph, typename VertexIndexMap,
              typename EdgeWeightMap, typename EdgeIndexMap>
    double
    minimum_cycle_mean(const Graph &g, VertexIndexMap vim,
                       EdgeWeightMap ewm, EdgeIndexMap eim,
                       std::vector<typename graph_traits<Graph>::edge_descriptor> *pcc = 0);
    template <typename FloatTraits, typename Graph, typename VertexIndexMap,
              typename EdgeWeightMap, typename EdgeIndexMap>

    typename FloatTraits::float_type
    minimum_cycle_mean(const Graph &g, VertexIndexMap vim,
                       EdgeWeightMap ewm, EdgeIndexMap eim,
                       std::vector<typename graph_traits<Graph>::edge_descriptor> *pcc = 0,
                       FloatTraits = FloatTraits());

### Where Defined

`boost/graph/howard_cycle_ratio.hpp`

### Parameters

IN: `FloatTraits`

> The `FloatTrats` encapsulates customizable limits-like information for floating point types. This type must provide an associated type, `value_type` for the floating point type. The default value is `boost::mcr_float<>`which has the following definition:

>      template <typename Float = double>
>      struct mcr_float {
>         typedef Float value_type;

>         static Float infinity()
>         { return (std::numeric_limits<value_type>::max)(); }

>         static Float epsilon()
>         { return Float(-0.005); }
>       };

> The value `FloatTraits::epsilon()` controls the "tolerance" of the algorithm. By increasing the absolute value of epsilon you may slightly decrease the execution time but there is a risk to skip a global optima. By decreasing the absolute value you may fall to the infinite loop. The best option is to leave this parameter unchanged.

IN: `const Graph& g`

> A weighted directed multigraph. The graph's type must be a model of VertexListGraph and IncidenceGraph

IN: `VertexIndexMap vim`

> Maps each vertex of the graph to a unique integer in the range [0, num_vertices(g)).

IN: `EdgeWeight1 ew1m`

> The W1 edge weight function.

IN: `EdgeWeight2 ew2m`

> The W2 edge weight function. Should be nonnegative. The actual limitation of the algorithm is the positivity of the total weight of each directed cycle of the graph.

OUT: `std::vector<typename boost::graph_traits<Graph>::edge_descriptor>* pcc`

> If non zero then one critical cycle will be stored in the std::vector. Default value is 0.

IN (only for maximum/minimal_cycle_mean()): `EdgeIndexMap eim`

> Maps each edge of the graph to a unique integer in the range [0, num_edges(g)).

> All property maps must be models of Readable Property Map

### Complexity

There is no known precise upper bound for the time complexity of the algorithm. Imperical time complexity is O(|E|), where the constant tends to be pretty small (about 20-30). Space complexity is equal to 7*|V| plus the space required to store a graph.

### Example

The program in libs/graph/example/cycle_ratio_example.cpp generates a random graph and computes its maximum cycle ratio.

Copyright © 2006-2009 | Dmitry Bufistov, Andrey Parfenov