/* -*- C++ -*-

  *

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

  *

  * Copyright (C) 2003-2006

  * 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

  * purpose.

  *

  */

 #ifndef LEMON_LP_BASE_H

 #define LEMON_LP_BASE_H

 #include<iostream>

 #include<vector>

 #include<map>

 #include<limits>

 #include<cmath>

 #include<lemon/error.h>

 #include<lemon/bits/invalid.h>

 #include<lemon/bits/utility.h>

 #include<lemon/bits/lp_id.h>

 ///\file

 ///\brief The interface of the LP solver interface.

 ///\ingroup lp_group

 namespace lemon {

   ///Common base class for LP solvers

   ///\todo Much more docs

   ///\ingroup lp_group

   class LpSolverBase {

   protected:

     _lp_bits::LpId rows;

     _lp_bits::LpId cols;

   public:

     ///Possible outcomes of an LP solving procedure

     enum SolveExitStatus {

       ///This means that the problem has been successfully solved: either

       ///an optimal solution has been found or infeasibility/unboundedness

       ///has been proved.

       SOLVED = 0,

       ///Any other case (including the case when some user specified

       ///limit has been exceeded)

       UNSOLVED = 1

     };

       ///\e

     enum SolutionStatus {

       ///Feasible solution hasn't been found (but may exist).

       ///\todo NOTFOUND might be a better name.

       ///

       UNDEFINED = 0,

       ///The problem has no feasible solution

       INFEASIBLE = 1,

       ///Feasible solution found

       FEASIBLE = 2,

       ///Optimal solution exists and found

       OPTIMAL = 3,

       ///The cost function is unbounded

       ///\todo Give a feasible solution and an infinite ray (and the

       ///corresponding bases)

       INFINITE = 4

     };

     ///\e The type of the investigated LP problem

     enum ProblemTypes {

       ///Primal-dual feasible

       PRIMAL_DUAL_FEASIBLE = 0,

       ///Primal feasible dual infeasible

       PRIMAL_FEASIBLE_DUAL_INFEASIBLE = 1,

       ///Primal infeasible dual feasible

       PRIMAL_INFEASIBLE_DUAL_FEASIBLE = 2,

       ///Primal-dual infeasible

       PRIMAL_DUAL_INFEASIBLE = 3,

       ///Could not determine so far

       UNKNOWN = 4

     };

     ///The floating point type used by the solver

     typedef double Value;

     ///The infinity constant

     static const Value INF;

     ///The not a number constant

     static const Value NaN;

     static inline bool isNaN(const Value& v) { return v!=v; }

     friend class Col;

     friend class ColIt;

     friend class Row;

     ///Refer to a column of the LP.

     ///This type is used to refer to a column of the LP.

     ///

     ///Its value remains valid and correct even after the addition or erase of

     ///other columns.

     ///

     ///\todo Document what can one do with a Col (INVALID, comparing,

     ///it is similar to Node/Edge)

     class Col {

     protected:

       int id;

       friend class LpSolverBase;

       friend class MipSolverBase;

       explicit Col(int _id) : id(_id) {}

     public:

       typedef Value ExprValue;

       typedef True LpSolverCol;

       Col() {}

       Col(const Invalid&) : id(-1) {}

       bool operator< (Col c) const  {return id< c.id;}

       bool operator> (Col c) const  {return id> c.id;}

       bool operator==(Col c) const  {return id==c.id;}

       bool operator!=(Col c) const  {return id!=c.id;}

     };

     class ColIt : public Col {

       const LpSolverBase *_lp;

     public:

       ColIt() {}

       ColIt(const LpSolverBase &lp) : _lp(&lp)

       {

         _lp->cols.firstFix(id);

       }

       ColIt(const Invalid&) : Col(INVALID) {}

       ColIt &operator++() 

       {

         _lp->cols.nextFix(id);

 	return *this;

       }

     };

     static int id(const Col& col) { return col.id; }

     ///Refer to a row of the LP.

     ///This type is used to refer to a row of the LP.

     ///

     ///Its value remains valid and correct even after the addition or erase of

     ///other rows.

     ///

     ///\todo Document what can one do with a Row (INVALID, comparing,

     ///it is similar to Node/Edge)

     class Row {

     protected:

       int id;

       friend class LpSolverBase;

       explicit Row(int _id) : id(_id) {}

     public:

       typedef Value ExprValue;

       typedef True LpSolverRow;

       Row() {}

       Row(const Invalid&) : id(-1) {}

       bool operator< (Row c) const  {return id< c.id;}

       bool operator> (Row c) const  {return id> c.id;}

       bool operator==(Row c) const  {return id==c.id;}

       bool operator!=(Row c) const  {return id!=c.id;} 

     };

     class RowIt : public Row {

       const LpSolverBase *_lp;

     public:

       RowIt() {}

       RowIt(const LpSolverBase &lp) : _lp(&lp)

       {

         _lp->rows.firstFix(id);

       }

       RowIt(const Invalid&) : Row(INVALID) {}

       RowIt &operator++() 

       {

         _lp->rows.nextFix(id);

 	return *this;

       }

     };

     static int id(const Row& row) { return row.id; }

   protected:

     int _lpId(const Col& col) const {

       return cols.floatingId(id(col));

     }

     int _lpId(const Row& row) const {

       return rows.floatingId(id(row));

     }

     Col _item(int id, Col) const {

       return Col(cols.fixId(id));

     }

     Row _item(int id, Row) const {

       return Row(rows.fixId(id));

     }

   public:

     ///Linear expression of variables and a constant component

     ///This data structure stores a linear expression of the variables

     ///(\ref Col "Col"s) and also has a constant component.

     ///

     ///There are several ways to access and modify the contents of this

     ///container.

     ///- Its it fully compatible with \c std::map<Col,double>, so for expamle

     ///if \c e is an Expr and \c v and \c w are of type \ref Col, then you can

     ///read and modify the coefficients like

     ///these.

     ///\code

     ///e[v]=5;

     ///e[v]+=12;

     ///e.erase(v);

     ///\endcode

     ///or you can also iterate through its elements.

     ///\code

     ///double s=0;

     ///for(LpSolverBase::Expr::iterator i=e.begin();i!=e.end();++i)

     ///  s+=i->second;

     ///\endcode

     ///(This code computes the sum of all coefficients).

     ///- Numbers (<tt>double</tt>'s)

     ///and variables (\ref Col "Col"s) directly convert to an

     ///\ref Expr and the usual linear operations are defined, so  

     ///\code

     ///v+w

     ///2*v-3.12*(v-w/2)+2

     ///v*2.1+(3*v+(v*12+w+6)*3)/2

     ///\endcode

     ///are valid \ref Expr "Expr"essions.

     ///The usual assignment operations are also defined.

     ///\code

     ///e=v+w;

     ///e+=2*v-3.12*(v-w/2)+2;

     ///e*=3.4;

     ///e/=5;

     ///\endcode

     ///- The constant member can be set and read by \ref constComp()

     ///\code

     ///e.constComp()=12;

     ///double c=e.constComp();

     ///\endcode

     ///

     ///\note \ref clear() not only sets all coefficients to 0 but also

     ///clears the constant components.

     ///

     ///\sa Constr

     ///

     class Expr : public std::map<Col,Value>

     {

     public:

       typedef LpSolverBase::Col Key; 

       typedef LpSolverBase::Value Value;

     protected:

       typedef std::map<Col,Value> Base;

       Value const_comp;

     public:

       typedef True IsLinExpression;

       ///\e

       Expr() : Base(), const_comp(0) { }

       ///\e

       Expr(const Key &v) : const_comp(0) {

 	Base::insert(std::make_pair(v, 1));

       }

       ///\e

       Expr(const Value &v) : const_comp(v) {}

       ///\e

       void set(const Key &v,const Value &c) {

 	Base::insert(std::make_pair(v, c));

       }

       ///\e

       Value &constComp() { return const_comp; }

       ///\e

       const Value &constComp() const { return const_comp; }

       ///Removes the components with zero coefficient.

       void simplify() {

 	for (Base::iterator i=Base::begin(); i!=Base::end();) {

 	  Base::iterator j=i;

 	  ++j;

 	  if ((*i).second==0) Base::erase(i);

 	  i=j;

 	}

       }

       void simplify() const {

         const_cast<Expr*>(this)->simplify();

       }

       ///Removes the coefficients closer to zero than \c tolerance.

       void simplify(double &tolerance) {

 	for (Base::iterator i=Base::begin(); i!=Base::end();) {

 	  Base::iterator j=i;

 	  ++j;

 	  if (std::fabs((*i).second)<tolerance) Base::erase(i);

 	  i=j;

 	}

       }

       ///Sets all coefficients and the constant component to 0.

       void clear() {

 	Base::clear();

 	const_comp=0;

       }

       ///\e

       Expr &operator+=(const Expr &e) {

 	for (Base::const_iterator j=e.begin(); j!=e.end(); ++j)

 	  (*this)[j->first]+=j->second;

 	const_comp+=e.const_comp;

 	return *this;

       }

       ///\e

       Expr &operator-=(const Expr &e) {

 	for (Base::const_iterator j=e.begin(); j!=e.end(); ++j)

 	  (*this)[j->first]-=j->second;

 	const_comp-=e.const_comp;

 	return *this;

       }

       ///\e

       Expr &operator*=(const Value &c) {

 	for (Base::iterator j=Base::begin(); j!=Base::end(); ++j)

 	  j->second*=c;

 	const_comp*=c;

 	return *this;

       }

       ///\e

       Expr &operator/=(const Value &c) {

 	for (Base::iterator j=Base::begin(); j!=Base::end(); ++j)

 	  j->second/=c;

 	const_comp/=c;

 	return *this;

       }

       //std::ostream &

       void prettyPrint(std::ostream &os) {

 	//std::fmtflags os.flags();

 	//os.setf(std::ios::showpos);

 	Base::iterator j=Base::begin();

 	if (j!=Base::end())

 	  os<<j->second<<"*x["<<id(j->first)<<"]";

 	++j;

 	for (; j!=Base::end(); ++j){

 	  if (j->second>=0)

 	    os<<"+";

 	  os<<j->second<<"*x["<<id(j->first)<<"]";

 	}

 	//Nem valami korrekt, de nem talaltam meg, hogy kell

 	//os.unsetf(std::ios::showpos);

 	//return os;

       }

     };

     ///Linear constraint

     ///This data stucture represents a linear constraint in the LP.

     ///Basically it is a linear expression with a lower or an upper bound

     ///(or both). These parts of the constraint can be obtained by the member

     ///functions \ref expr(), \ref lowerBound() and \ref upperBound(),

     ///respectively.

     ///There are two ways to construct a constraint.

     ///- You can set the linear expression and the bounds directly

     ///  by the functions above.

     ///- The operators <tt>\<=</tt>, <tt>==</tt> and  <tt>\>=</tt>

     ///  are defined between expressions, or even between constraints whenever

     ///  it makes sense. Therefore if \c e and \c f are linear expressions and

     ///  \c s and \c t are numbers, then the followings are valid expressions

     ///  and thus they can be used directly e.g. in \ref addRow() whenever

     ///  it makes sense.

     ///\code

     ///  e<=s

     ///  e<=f

     ///  e==f

     ///  s<=e<=t

     ///  e>=t

     ///\endcode

     ///\warning The validity of a constraint is checked only at run time, so

     ///e.g. \ref addRow(<tt>x[1]\<=x[2]<=5</tt>) will compile, but will throw a

     ///\ref LogicError exception.

     class Constr

     {

     public:

       typedef LpSolverBase::Expr Expr;

       typedef Expr::Key Key;

       typedef Expr::Value Value;

     protected:

       Expr _expr;

       Value _lb,_ub;

     public:

       ///\e

       Constr() : _expr(), _lb(NaN), _ub(NaN) {}

       ///\e

       Constr(Value lb,const Expr &e,Value ub) :

 	_expr(e), _lb(lb), _ub(ub) {}

       ///\e

       Constr(const Expr &e,Value ub) : 

 	_expr(e), _lb(NaN), _ub(ub) {}

       ///\e

       Constr(Value lb,const Expr &e) :

 	_expr(e), _lb(lb), _ub(NaN) {}

       ///\e

       Constr(const Expr &e) : 

 	_expr(e), _lb(NaN), _ub(NaN) {}

       ///\e

       void clear() 

       {

 	_expr.clear();

 	_lb=_ub=NaN;

       }

       ///Reference to the linear expression 

       Expr &expr() { return _expr; }

       ///Cont reference to the linear expression 

       const Expr &expr() const { return _expr; }

       ///Reference to the lower bound.

       ///\return

       ///- \ref INF "INF": the constraint is lower unbounded.

       ///- \ref NaN "NaN": lower bound has not been set.

       ///- finite number: the lower bound

       Value &lowerBound() { return _lb; }

       ///The const version of \ref lowerBound()

       const Value &lowerBound() const { return _lb; }

       ///Reference to the upper bound.

       ///\return

       ///- \ref INF "INF": the constraint is upper unbounded.

       ///- \ref NaN "NaN": upper bound has not been set.

       ///- finite number: the upper bound

       Value &upperBound() { return _ub; }

       ///The const version of \ref upperBound()

       const Value &upperBound() const { return _ub; }

       ///Is the constraint lower bounded?

       bool lowerBounded() const { 

 	using namespace std;

 	return finite(_lb);

       }

       ///Is the constraint upper bounded?

       bool upperBounded() const {

 	using namespace std;

 	return finite(_ub);

       }

       void prettyPrint(std::ostream &os) {

 	if (_lb==-LpSolverBase::INF||isNaN(_lb))

 	  os<<"-infty<=";

 	else

 	  os<<_lb<<"<=";

 	_expr.prettyPrint(os);

 	if (_ub==LpSolverBase::INF)

 	  os<<"<=infty";

 	else

 	  os<<"<="<<_ub;

 	//return os;

       }

     };

     ///Linear expression of rows

     ///This data structure represents a column of the matrix,

     ///thas is it strores a linear expression of the dual variables

     ///(\ref Row "Row"s).

     ///

     ///There are several ways to access and modify the contents of this

     ///container.

     ///- Its it fully compatible with \c std::map<Row,double>, so for expamle

     ///if \c e is an DualExpr and \c v

     ///and \c w are of type \ref Row, then you can

     ///read and modify the coefficients like

     ///these.

     ///\code

     ///e[v]=5;

     ///e[v]+=12;

     ///e.erase(v);

     ///\endcode

     ///or you can also iterate through its elements.

     ///\code

     ///double s=0;

     ///for(LpSolverBase::DualExpr::iterator i=e.begin();i!=e.end();++i)

     ///  s+=i->second;

     ///\endcode

     ///(This code computes the sum of all coefficients).

     ///- Numbers (<tt>double</tt>'s)

     ///and variables (\ref Row "Row"s) directly convert to an

     ///\ref DualExpr and the usual linear operations are defined, so

     ///\code

     ///v+w

     ///2*v-3.12*(v-w/2)

     ///v*2.1+(3*v+(v*12+w)*3)/2

     ///\endcode

     ///are valid \ref DualExpr "DualExpr"essions.

     ///The usual assignment operations are also defined.

     ///\code

     ///e=v+w;

     ///e+=2*v-3.12*(v-w/2);

     ///e*=3.4;

     ///e/=5;

     ///\endcode

     ///

     ///\sa Expr

     ///

     class DualExpr : public std::map<Row,Value>

     {

     public:

       typedef LpSolverBase::Row Key; 

       typedef LpSolverBase::Value Value;

     protected:

       typedef std::map<Row,Value> Base;

     public:

       typedef True IsLinExpression;

       ///\e

       DualExpr() : Base() { }

       ///\e

       DualExpr(const Key &v) {

 	Base::insert(std::make_pair(v, 1));

       }

       ///\e

       void set(const Key &v,const Value &c) {

 	Base::insert(std::make_pair(v, c));

       }

       ///Removes the components with zero coefficient.

       void simplify() {

 	for (Base::iterator i=Base::begin(); i!=Base::end();) {

 	  Base::iterator j=i;

 	  ++j;

 	  if ((*i).second==0) Base::erase(i);

 	  i=j;

 	}

       }

       void simplify() const {

         const_cast<DualExpr*>(this)->simplify();

       }

       ///Removes the coefficients closer to zero than \c tolerance.

       void simplify(double &tolerance) {

 	for (Base::iterator i=Base::begin(); i!=Base::end();) {

 	  Base::iterator j=i;

 	  ++j;

 	  if (std::fabs((*i).second)<tolerance) Base::erase(i);

 	  i=j;

 	}

       }

       ///Sets all coefficients to 0.

       void clear() {

 	Base::clear();

       }

       ///\e

       DualExpr &operator+=(const DualExpr &e) {

 	for (Base::const_iterator j=e.begin(); j!=e.end(); ++j)

 	  (*this)[j->first]+=j->second;

 	return *this;

       }

       ///\e

       DualExpr &operator-=(const DualExpr &e) {

 	for (Base::const_iterator j=e.begin(); j!=e.end(); ++j)

 	  (*this)[j->first]-=j->second;

 	return *this;

       }

       ///\e

       DualExpr &operator*=(const Value &c) {

 	for (Base::iterator j=Base::begin(); j!=Base::end(); ++j)

 	  j->second*=c;

 	return *this;

       }

       ///\e

       DualExpr &operator/=(const Value &c) {

 	for (Base::iterator j=Base::begin(); j!=Base::end(); ++j)

 	  j->second/=c;

 	return *this;

       }

     };

   private:

     template <typename _Expr>

     class MappedOutputIterator {

     public:

       typedef std::insert_iterator<_Expr> Base;

       typedef std::output_iterator_tag iterator_category;

       typedef void difference_type;

       typedef void value_type;

       typedef void reference;

       typedef void pointer;

       MappedOutputIterator(const Base& _base, const LpSolverBase& _lp) 

         : base(_base), lp(_lp) {}

       MappedOutputIterator& operator*() {

         return *this;

       }

       MappedOutputIterator& operator=(const std::pair<int, Value>& value) {

         *base = std::make_pair(lp._item(value.first, typename _Expr::Key()), 

                                value.second);

         return *this;

       }

       MappedOutputIterator& operator++() {

         ++base;

         return *this;

       }

       MappedOutputIterator operator++(int) {

         MappedOutputIterator tmp(*this);

         ++base;

         return tmp;

       }

       bool operator==(const MappedOutputIterator& it) const {

         return base == it.base;

       }

       bool operator!=(const MappedOutputIterator& it) const {

         return base != it.base;

       }

     private:

       Base base;

       const LpSolverBase& lp;

     };

     template <typename Expr>

     class MappedInputIterator {

     public:

       typedef typename Expr::const_iterator Base;

       typedef typename Base::iterator_category iterator_category;

       typedef typename Base::difference_type difference_type;

       typedef const std::pair<int, Value> value_type;

       typedef value_type reference;

       class pointer {

       public:

         pointer(value_type& _value) : value(_value) {}

         value_type* operator->() { return &value; }

       private:

         value_type value;

       };

       MappedInputIterator(const Base& _base, const LpSolverBase& _lp) 

         : base(_base), lp(_lp) {}

       reference operator*() {

         return std::make_pair(lp._lpId(base->first), base->second);

       }

       pointer operator->() {

         return pointer(operator*());

       }

       MappedInputIterator& operator++() {

         ++base;

         return *this;

       }

       MappedInputIterator operator++(int) {

         MappedInputIterator tmp(*this);

         ++base;

         return tmp;

       }

       bool operator==(const MappedInputIterator& it) const {

         return base == it.base;

       }

       bool operator!=(const MappedInputIterator& it) const {

         return base != it.base;

       }

     private:

       Base base;

       const LpSolverBase& lp;

     };

   protected:

     /// STL compatible iterator for lp col

     typedef MappedInputIterator<Expr> ConstRowIterator;

     /// STL compatible iterator for lp row

     typedef MappedInputIterator<DualExpr> ConstColIterator;

     /// STL compatible iterator for lp col

     typedef MappedOutputIterator<Expr> RowIterator;

     /// STL compatible iterator for lp row

     typedef MappedOutputIterator<DualExpr> ColIterator;

     //Abstract virtual functions

     virtual LpSolverBase &_newLp() = 0;

     virtual LpSolverBase &_copyLp(){

       ///\todo This should be implemented here, too, when we have

       ///problem retrieving routines. It can be overriden.

       //Starting:

       LpSolverBase & newlp(_newLp());

       return newlp;

       //return *(LpSolverBase*)0;

     };

     virtual int _addCol() = 0;

     virtual int _addRow() = 0; 

     virtual void _eraseCol(int col) = 0;

     virtual void _eraseRow(int row) = 0;

     virtual void _getColName(int col, std::string & name) const = 0;

     virtual void _setColName(int col, const std::string & name) = 0;

     virtual int _colByName(const std::string& name) const = 0;

     virtual void _setRowCoeffs(int i, ConstRowIterator b, 

                                ConstRowIterator e) = 0;

     virtual void _getRowCoeffs(int i, RowIterator b) const = 0;

     virtual void _setColCoeffs(int i, ConstColIterator b, 

                                ConstColIterator e) = 0;

     virtual void _getColCoeffs(int i, ColIterator b) const = 0;

     virtual void _setCoeff(int row, int col, Value value) = 0;

     virtual Value _getCoeff(int row, int col) const = 0;

     virtual void _setColLowerBound(int i, Value value) = 0;

     virtual Value _getColLowerBound(int i) const = 0;

     virtual void _setColUpperBound(int i, Value value) = 0;

     virtual Value _getColUpperBound(int i) const = 0;

     virtual void _setRowBounds(int i, Value lower, Value upper) = 0;

     virtual void _getRowBounds(int i, Value &lower, Value &upper) const = 0;

     virtual void _setObjCoeff(int i, Value obj_coef) = 0;

     virtual Value _getObjCoeff(int i) const = 0;

     virtual void _clearObj()=0;

     virtual SolveExitStatus _solve() = 0;

     virtual Value _getPrimal(int i) const = 0;

     virtual Value _getDual(int i) const = 0;

     virtual Value _getPrimalValue() const = 0;

     virtual bool _isBasicCol(int i) const = 0;

     virtual SolutionStatus _getPrimalStatus() const = 0;

     virtual SolutionStatus _getDualStatus() const = 0;

     virtual ProblemTypes _getProblemType() const = 0;

     virtual void _setMax() = 0;

     virtual void _setMin() = 0;

     virtual bool _isMax() const = 0;

     //Own protected stuff

     //Constant component of the objective function

     Value obj_const_comp;

   public:

     ///\e

     LpSolverBase() : obj_const_comp(0) {}

     ///\e

     virtual ~LpSolverBase() {}

     ///Creates a new LP problem

     LpSolverBase &newLp() {return _newLp();}

     ///Makes a copy of the LP problem

     LpSolverBase &copyLp() {return _copyLp();}

     ///\name Build up and modify the LP

     ///@{

     ///Add a new empty column (i.e a new variable) to the LP

     Col addCol() { Col c; _addCol(); c.id = cols.addId(); return c;}

     ///\brief Adds several new columns

     ///(i.e a variables) at once

     ///

     ///This magic function takes a container as its argument

     ///and fills its elements

     ///with new columns (i.e. variables)

     ///\param t can be

     ///- a standard STL compatible iterable container with

     ///\ref Col as its \c values_type

     ///like

     ///\code

     ///std::vector<LpSolverBase::Col>

     ///std::list<LpSolverBase::Col>

     ///\endcode

     ///- a standard STL compatible iterable container with

     ///\ref Col as its \c mapped_type

     ///like

     ///\code

     ///std::map<AnyType,LpSolverBase::Col>

     ///\endcode

     ///- an iterable lemon \ref concepts::WriteMap "write map" like 

     ///\code

     ///ListGraph::NodeMap<LpSolverBase::Col>

     ///ListGraph::EdgeMap<LpSolverBase::Col>

     ///\endcode

     ///\return The number of the created column.

 #ifdef DOXYGEN

     template<class T>

     int addColSet(T &t) { return 0;} 

 #else

     template<class T>

     typename enable_if<typename T::value_type::LpSolverCol,int>::type

     addColSet(T &t,dummy<0> = 0) {

       int s=0;

       for(typename T::iterator i=t.begin();i!=t.end();++i) {*i=addCol();s++;}

       return s;

     }

     template<class T>

     typename enable_if<typename T::value_type::second_type::LpSolverCol,

 		       int>::type

     addColSet(T &t,dummy<1> = 1) { 

       int s=0;

       for(typename T::iterator i=t.begin();i!=t.end();++i) {

 	i->second=addCol();

 	s++;

       }

       return s;

     }

     template<class T>

     typename enable_if<typename T::MapIt::Value::LpSolverCol,

 		       int>::type

     addColSet(T &t,dummy<2> = 2) { 

       int s=0;

       for(typename T::MapIt i(t); i!=INVALID; ++i)

 	{

 	  i.set(addCol());

 	  s++;

 	}

       return s;

     }

 #endif

     ///Set a column (i.e a dual constraint) of the LP

     ///\param c is the column to be modified

     ///\param e is a dual linear expression (see \ref DualExpr)

     ///a better one.

     void col(Col c,const DualExpr &e) {

       e.simplify();

       _setColCoeffs(_lpId(c), ConstColIterator(e.begin(), *this), 

                     ConstColIterator(e.end(), *this));

     }

     ///Get a column (i.e a dual constraint) of the LP

     ///\param r is the column to get

     ///\return the dual expression associated to the column

     DualExpr col(Col c) const {

       DualExpr e;

       _getColCoeffs(_lpId(c), ColIterator(std::inserter(e, e.end()), *this));

       return e;

     }

     ///Add a new column to the LP

     ///\param e is a dual linear expression (see \ref DualExpr)

     ///\param obj is the corresponding component of the objective

     ///function. It is 0 by default.

     ///\return The created column.

     Col addCol(const DualExpr &e, Value obj=0) {

       Col c=addCol();

       col(c,e);

       objCoeff(c,obj);

       return c;

     }

     ///Add a new empty row (i.e a new constraint) to the LP

     ///This function adds a new empty row (i.e a new constraint) to the LP.

     ///\return The created row

     Row addRow() { Row r; _addRow(); r.id = rows.addId(); return r;}

     ///\brief Add several new rows

     ///(i.e a constraints) at once

     ///

     ///This magic function takes a container as its argument

     ///and fills its elements

     ///with new row (i.e. variables)

     ///\param t can be

     ///- a standard STL compatible iterable container with

     ///\ref Row as its \c values_type

     ///like

     ///\code

     ///std::vector<LpSolverBase::Row>

     ///std::list<LpSolverBase::Row>

     ///\endcode

     ///- a standard STL compatible iterable container with

     ///\ref Row as its \c mapped_type

     ///like

     ///\code

     ///std::map<AnyType,LpSolverBase::Row>

     ///\endcode

     ///- an iterable lemon \ref concepts::WriteMap "write map" like 

     ///\code

     ///ListGraph::NodeMap<LpSolverBase::Row>

     ///ListGraph::EdgeMap<LpSolverBase::Row>

     ///\endcode

     ///\return The number of rows created.

 #ifdef DOXYGEN

     template<class T>

     int addRowSet(T &t) { return 0;} 

 #else

     template<class T>

     typename enable_if<typename T::value_type::LpSolverRow,int>::type

     addRowSet(T &t,dummy<0> = 0) {

       int s=0;

       for(typename T::iterator i=t.begin();i!=t.end();++i) {*i=addRow();s++;}

       return s;

     }

     template<class T>

     typename enable_if<typename T::value_type::second_type::LpSolverRow,

 		       int>::type

     addRowSet(T &t,dummy<1> = 1) { 

       int s=0;

       for(typename T::iterator i=t.begin();i!=t.end();++i) {

 	i->second=addRow();

 	s++;

       }

       return s;

     }

     template<class T>

     typename enable_if<typename T::MapIt::Value::LpSolverRow,

 		       int>::type

     addRowSet(T &t,dummy<2> = 2) { 

       int s=0;

       for(typename T::MapIt i(t); i!=INVALID; ++i)

 	{

 	  i.set(addRow());

 	  s++;

 	}

       return s;

     }

 #endif

     ///Set a row (i.e a constraint) of the LP

     ///\param r is the row to be modified

     ///\param l is lower bound (-\ref INF means no bound)

     ///\param e is a linear expression (see \ref Expr)

     ///\param u is the upper bound (\ref INF means no bound)

     ///\bug This is a temporary function. The interface will change to

     ///a better one.

     ///\todo Option to control whether a constraint with a single variable is

     ///added or not.

     void row(Row r, Value l, const Expr &e, Value u) {

       e.simplify();

       _setRowCoeffs(_lpId(r), ConstRowIterator(e.begin(), *this),

                     ConstRowIterator(e.end(), *this));

       _setRowBounds(_lpId(r),l-e.constComp(),u-e.constComp());

     }

     ///Set a row (i.e a constraint) of the LP

     ///\param r is the row to be modified

     ///\param c is a linear expression (see \ref Constr)

     void row(Row r, const Constr &c) {

       row(r, c.lowerBounded()?c.lowerBound():-INF,

           c.expr(), c.upperBounded()?c.upperBound():INF);

     }

     ///Get a row (i.e a constraint) of the LP

     ///\param r is the row to get

     ///\return the expression associated to the row

     Expr row(Row r) const {

       Expr e;

       _getRowCoeffs(_lpId(r), RowIterator(std::inserter(e, e.end()), *this));

       return e;

     }

     ///Add a new row (i.e a new constraint) to the LP

     ///\param l is the lower bound (-\ref INF means no bound)

     ///\param e is a linear expression (see \ref Expr)

     ///\param u is the upper bound (\ref INF means no bound)

     ///\return The created row.

     ///\bug This is a temporary function. The interface will change to

     ///a better one.

     Row addRow(Value l,const Expr &e, Value u) {

       Row r=addRow();

       row(r,l,e,u);

       return r;

     }

     ///Add a new row (i.e a new constraint) to the LP

     ///\param c is a linear expression (see \ref Constr)

     ///\return The created row.

     Row addRow(const Constr &c) {

       Row r=addRow();

       row(r,c);

       return r;

     }

     ///Erase a coloumn (i.e a variable) from the LP

     ///\param c is the coloumn to be deleted

     ///\todo Please check this

     void eraseCol(Col c) {

       _eraseCol(_lpId(c));

       cols.eraseId(c.id);

     }

     ///Erase a  row (i.e a constraint) from the LP

     ///\param r is the row to be deleted

     ///\todo Please check this

     void eraseRow(Row r) {

       _eraseRow(_lpId(r));

       rows.eraseId(r.id);

     }

     /// Get the name of a column

     ///\param c is the coresponding coloumn 

     ///\return The name of the colunm

     std::string colName(Col c) const {

       std::string name;

       _getColName(_lpId(c), name);

       return name;

     }

     /// Set the name of a column

     ///\param c is the coresponding coloumn 

     ///\param name The name to be given

     void colName(Col c, const std::string& name) {

       _setColName(_lpId(c), name);

     }

     /// Get the column by its name

     ///\param name The name of the column

     ///\return the proper column or \c INVALID

     Col colByName(const std::string& name) const {

       int k = _colByName(name);

       return k != -1 ? Col(cols.fixId(k)) : Col(INVALID);

     }

     /// Set an element of the coefficient matrix of the LP

     ///\param r is the row of the element to be modified

     ///\param c is the coloumn of the element to be modified

     ///\param val is the new value of the coefficient

     void coeff(Row r, Col c, Value val) {

       _setCoeff(_lpId(r),_lpId(c), val);

     }

     /// Get an element of the coefficient matrix of the LP

     ///\param r is the row of the element in question

     ///\param c is the coloumn of the element in question

     ///\return the corresponding coefficient

     Value coeff(Row r, Col c) const {

       return _getCoeff(_lpId(r),_lpId(c));

     }

     /// Set the lower bound of a column (i.e a variable)

     /// The lower bound of a variable (column) has to be given by an 

     /// extended number of type Value, i.e. a finite number of type 

     /// Value or -\ref INF.

     void colLowerBound(Col c, Value value) {

       _setColLowerBound(_lpId(c),value);

     }

     /// Get the lower bound of a column (i.e a variable)

     /// This function returns the lower bound for column (variable) \t c

     /// (this might be -\ref INF as well).  

     ///\return The lower bound for coloumn \t c

     Value colLowerBound(Col c) const {

       return _getColLowerBound(_lpId(c));

     }

     ///\brief Set the lower bound of  several columns

     ///(i.e a variables) at once

     ///

     ///This magic function takes a container as its argument

     ///and applies the function on all of its elements.

     /// The lower bound of a variable (column) has to be given by an 

     /// extended number of type Value, i.e. a finite number of type 

     /// Value or -\ref INF.

 #ifdef DOXYGEN

     template<class T>

     void colLowerBound(T &t, Value value) { return 0;} 

 #else

     template<class T>

     typename enable_if<typename T::value_type::LpSolverCol,void>::type

     colLowerBound(T &t, Value value,dummy<0> = 0) {

       for(typename T::iterator i=t.begin();i!=t.end();++i) {

 	colLowerBound(*i, value);

       }

     }

     template<class T>

     typename enable_if<typename T::value_type::second_type::LpSolverCol,

 		       void>::type

     colLowerBound(T &t, Value value,dummy<1> = 1) { 

       for(typename T::iterator i=t.begin();i!=t.end();++i) {

 	colLowerBound(i->second, value);

       }

     }

     template<class T>

     typename enable_if<typename T::MapIt::Value::LpSolverCol,

 		       void>::type

     colLowerBound(T &t, Value value,dummy<2> = 2) { 

       for(typename T::MapIt i(t); i!=INVALID; ++i){

 	colLowerBound(*i, value);

       }

     }

 #endif

     /// Set the upper bound of a column (i.e a variable)

     /// The upper bound of a variable (column) has to be given by an 

     /// extended number of type Value, i.e. a finite number of type 

     /// Value or \ref INF.

     void colUpperBound(Col c, Value value) {

       _setColUpperBound(_lpId(c),value);

     };

     /// Get the upper bound of a column (i.e a variable)

     /// This function returns the upper bound for column (variable) \t c

     /// (this might be \ref INF as well).  

     ///\return The upper bound for coloumn \t c

     Value colUpperBound(Col c) const {

       return _getColUpperBound(_lpId(c));

     }

     ///\brief Set the upper bound of  several columns

     ///(i.e a variables) at once

     ///

     ///This magic function takes a container as its argument

     ///and applies the function on all of its elements.

     /// The upper bound of a variable (column) has to be given by an 

     /// extended number of type Value, i.e. a finite number of type 

     /// Value or \ref INF.

 #ifdef DOXYGEN

     template<class T>

     void colUpperBound(T &t, Value value) { return 0;} 

 #else

     template<class T>

     typename enable_if<typename T::value_type::LpSolverCol,void>::type

     colUpperBound(T &t, Value value,dummy<0> = 0) {

       for(typename T::iterator i=t.begin();i!=t.end();++i) {

 	colUpperBound(*i, value);

       }

     }

     template<class T>

     typename enable_if<typename T::value_type::second_type::LpSolverCol,

 		       void>::type

     colUpperBound(T &t, Value value,dummy<1> = 1) { 

       for(typename T::iterator i=t.begin();i!=t.end();++i) {

 	colUpperBound(i->second, value);

       }

     }

     template<class T>

     typename enable_if<typename T::MapIt::Value::LpSolverCol,

 		       void>::type

     colUpperBound(T &t, Value value,dummy<2> = 2) { 

       for(typename T::MapIt i(t); i!=INVALID; ++i){

 	colUpperBound(*i, value);

       }

     }

 #endif

     /// Set the lower and the upper bounds of a column (i.e a variable)

     /// The lower and the upper bounds of

     /// a variable (column) have to be given by an 

     /// extended number of type Value, i.e. a finite number of type 

     /// Value, -\ref INF or \ref INF.

     void colBounds(Col c, Value lower, Value upper) {

       _setColLowerBound(_lpId(c),lower);

       _setColUpperBound(_lpId(c),upper);

     }

     ///\brief Set the lower and the upper bound of several columns

     ///(i.e a variables) at once

     ///

     ///This magic function takes a container as its argument

     ///and applies the function on all of its elements.

     /// The lower and the upper bounds of

     /// a variable (column) have to be given by an 

     /// extended number of type Value, i.e. a finite number of type 

     /// Value, -\ref INF or \ref INF.

 #ifdef DOXYGEN

     template<class T>

     void colBounds(T &t, Value lower, Value upper) { return 0;} 

 #else

     template<class T>

     typename enable_if<typename T::value_type::LpSolverCol,void>::type

     colBounds(T &t, Value lower, Value upper,dummy<0> = 0) {

       for(typename T::iterator i=t.begin();i!=t.end();++i) {

 	colBounds(*i, lower, upper);

       }

     }

     template<class T>

     typename enable_if<typename T::value_type::second_type::LpSolverCol,

 		       void>::type

     colBounds(T &t, Value lower, Value upper,dummy<1> = 1) { 

       for(typename T::iterator i=t.begin();i!=t.end();++i) {

 	colBounds(i->second, lower, upper);

       }

     }

     template<class T>

     typename enable_if<typename T::MapIt::Value::LpSolverCol,

 		       void>::type

     colBounds(T &t, Value lower, Value upper,dummy<2> = 2) { 

       for(typename T::MapIt i(t); i!=INVALID; ++i){

 	colBounds(*i, lower, upper);

       }

     }

 #endif

     /// Set the lower and the upper bounds of a row (i.e a constraint)

     /// The lower and the upper bound of a constraint (row) have to be

     /// given by an extended number of type Value, i.e. a finite

     /// number of type Value, -\ref INF or \ref INF. There is no

     /// separate function for the lower and the upper bound because

     /// that would have been hard to implement for CPLEX.

     void rowBounds(Row c, Value lower, Value upper) {

       _setRowBounds(_lpId(c),lower, upper);

     }

     /// Get the lower and the upper bounds of a row (i.e a constraint)

     /// The lower and the upper bound of

     /// a constraint (row) are  

     /// extended numbers of type Value, i.e.  finite numbers of type 

     /// Value, -\ref INF or \ref INF. 

     /// \todo There is no separate function for the 

     /// lower and the upper bound because we had problems with the 

     /// implementation of the setting functions for CPLEX:  

     /// check out whether this can be done for these functions.

     void getRowBounds(Row c, Value &lower, Value &upper) const {

       _getRowBounds(_lpId(c),lower, upper);

     }

     ///Set an element of the objective function

     void objCoeff(Col c, Value v) {_setObjCoeff(_lpId(c),v); };

     ///Get an element of the objective function

     Value objCoeff(Col c) const { return _getObjCoeff(_lpId(c)); };

     ///Set the objective function

     ///\param e is a linear expression of type \ref Expr.

     void obj(Expr e) {

       _clearObj();

       for (Expr::iterator i=e.begin(); i!=e.end(); ++i)

 	objCoeff((*i).first,(*i).second);

       obj_const_comp=e.constComp();

     }

     ///Get the objective function

     ///\return the objective function as a linear expression of type \ref Expr.

     Expr obj() const {

       Expr e;

       for (ColIt it(*this); it != INVALID; ++it) {

         double c = objCoeff(it);

         if (c != 0.0) {

           e.insert(std::make_pair(it, c));

         }

       }

       return e;

     }

     ///Maximize

     void max() { _setMax(); }

     ///Minimize

     void min() { _setMin(); }

     ///Query function: is this a maximization problem?

     bool isMax() const {return _isMax(); }

     ///Query function: is this a minimization problem?

     bool isMin() const {return !isMax(); }

     ///@}

     ///\name Solve the LP

     ///@{

     ///\e Solve the LP problem at hand

     ///

     ///\return The result of the optimization procedure. Possible 

     ///values and their meanings can be found in the documentation of 

     ///\ref SolveExitStatus.

     ///

     ///\todo Which method is used to solve the problem

     SolveExitStatus solve() { return _solve(); }

     ///@}

     ///\name Obtain the solution

     ///@{

     /// The status of the primal problem (the original LP problem)

     SolutionStatus primalStatus() const {

       return _getPrimalStatus();

     }

     /// The status of the dual (of the original LP) problem 

     SolutionStatus dualStatus() const {

       return _getDualStatus();

     }

     ///The type of the original LP problem

     ProblemTypes problemType() const {

       return _getProblemType();

     }

     ///\e

     Value primal(Col c) const { return _getPrimal(_lpId(c)); }

     ///\e

     Value dual(Row r) const { return _getDual(_lpId(r)); }

     ///\e

     bool isBasicCol(Col c) const { return _isBasicCol(_lpId(c)); }

     ///\e

     ///\return

     ///- \ref INF or -\ref INF means either infeasibility or unboundedness

     /// of the primal problem, depending on whether we minimize or maximize.

     ///- \ref NaN if no primal solution is found.

     ///- The (finite) objective value if an optimal solution is found.

     Value primalValue() const { return _getPrimalValue()+obj_const_comp;}

     ///@}

   };  

   /// \ingroup lp_group

   ///

   /// \brief Common base class for MIP solvers

   /// \todo Much more docs

   class MipSolverBase : virtual public LpSolverBase{

   public:

     ///Possible variable (coloumn) types (e.g. real, integer, binary etc.)

     enum ColTypes {

       ///Continuous variable

       REAL = 0,

       ///Integer variable

       ///Unfortunately, cplex 7.5 somewhere writes something like

       ///#define INTEGER 'I'

       INT = 1

       ///\todo No support for other types yet.

     };

     ///Sets the type of the given coloumn to the given type

     ///

     ///Sets the type of the given coloumn to the given type.

     void colType(Col c, ColTypes col_type) {

       _colType(_lpId(c),col_type);

     }

     ///Gives back the type of the column.

     ///

     ///Gives back the type of the column.

     ColTypes colType(Col c) const {

       return _colType(_lpId(c));

     }

     ///Sets the type of the given Col to integer or remove that property.

     ///

     ///Sets the type of the given Col to integer or remove that property.

     void integer(Col c, bool enable) {

       if (enable)

 	colType(c,INT);

       else

 	colType(c,REAL);

     }

     ///Gives back whether the type of the column is integer or not.

     ///

     ///Gives back the type of the column.

     ///\return true if the column has integer type and false if not.

     bool integer(Col c) const {

       return (colType(c)==INT);

     }

     /// The status of the MIP problem

     SolutionStatus mipStatus() const {

       return _getMipStatus();

     }

   protected:

     virtual ColTypes _colType(int col) const = 0;

     virtual void _colType(int col, ColTypes col_type) = 0;

     virtual SolutionStatus _getMipStatus() const = 0;

   };

   ///\relates LpSolverBase::Expr

   ///

   inline LpSolverBase::Expr operator+(const LpSolverBase::Expr &a,

 				      const LpSolverBase::Expr &b) 

   {

     LpSolverBase::Expr tmp(a);

     tmp+=b;

     return tmp;

   }

   ///\e

   ///\relates LpSolverBase::Expr

   ///

   inline LpSolverBase::Expr operator-(const LpSolverBase::Expr &a,

 				      const LpSolverBase::Expr &b) 

   {

     LpSolverBase::Expr tmp(a);

     tmp-=b;

     return tmp;

   }

   ///\e

   ///\relates LpSolverBase::Expr

   ///

   inline LpSolverBase::Expr operator*(const LpSolverBase::Expr &a,

 				      const LpSolverBase::Value &b) 

   {

     LpSolverBase::Expr tmp(a);

     tmp*=b;

     return tmp;

   }

   ///\e

   ///\relates LpSolverBase::Expr

   ///

   inline LpSolverBase::Expr operator*(const LpSolverBase::Value &a,

 				      const LpSolverBase::Expr &b) 

   {

     LpSolverBase::Expr tmp(b);

     tmp*=a;

     return tmp;

   }

   ///\e

   ///\relates LpSolverBase::Expr

   ///

   inline LpSolverBase::Expr operator/(const LpSolverBase::Expr &a,

 				      const LpSolverBase::Value &b) 

   {

     LpSolverBase::Expr tmp(a);

     tmp/=b;

     return tmp;

   }

   ///\e

   ///\relates LpSolverBase::Constr

   ///

   inline LpSolverBase::Constr operator<=(const LpSolverBase::Expr &e,

 					 const LpSolverBase::Expr &f) 

   {

     return LpSolverBase::Constr(-LpSolverBase::INF,e-f,0);

   }

   ///\e

   ///\relates LpSolverBase::Constr

   ///

   inline LpSolverBase::Constr operator<=(const LpSolverBase::Value &e,

 					 const LpSolverBase::Expr &f) 

   {

     return LpSolverBase::Constr(e,f);

   }

   ///\e

   ///\relates LpSolverBase::Constr

   ///

   inline LpSolverBase::Constr operator<=(const LpSolverBase::Expr &e,

 					 const LpSolverBase::Value &f) 

   {

     return LpSolverBase::Constr(e,f);

   }

   ///\e

   ///\relates LpSolverBase::Constr

   ///

   inline LpSolverBase::Constr operator>=(const LpSolverBase::Expr &e,

 					 const LpSolverBase::Expr &f) 

   {

     return LpSolverBase::Constr(-LpSolverBase::INF,f-e,0);

   }

   ///\e

   ///\relates LpSolverBase::Constr

   ///

   inline LpSolverBase::Constr operator>=(const LpSolverBase::Value &e,

 					 const LpSolverBase::Expr &f) 

   {

     return LpSolverBase::Constr(f,e);

   }

   ///\e

   ///\relates LpSolverBase::Constr

   ///

   inline LpSolverBase::Constr operator>=(const LpSolverBase::Expr &e,

 					 const LpSolverBase::Value &f) 

   {

     return LpSolverBase::Constr(f,e);

   }

   ///\e

   ///\relates LpSolverBase::Constr

   ///

   inline LpSolverBase::Constr operator==(const LpSolverBase::Expr &e,

 					 const LpSolverBase::Value &f) 

   {

     return LpSolverBase::Constr(f,e,f);

   }

   ///\e

   ///\relates LpSolverBase::Constr

   ///

   inline LpSolverBase::Constr operator==(const LpSolverBase::Expr &e,

 					 const LpSolverBase::Expr &f) 

   {

     return LpSolverBase::Constr(0,e-f,0);

   }

   ///\e

   ///\relates LpSolverBase::Constr

   ///

   inline LpSolverBase::Constr operator<=(const LpSolverBase::Value &n,

 					 const LpSolverBase::Constr&c) 

   {

     LpSolverBase::Constr tmp(c);

     ///\todo Create an own exception type.

     if(!LpSolverBase::isNaN(tmp.lowerBound())) throw LogicError();

     else tmp.lowerBound()=n;

     return tmp;

   }

   ///\e

   ///\relates LpSolverBase::Constr

   ///

   inline LpSolverBase::Constr operator<=(const LpSolverBase::Constr& c,

 					 const LpSolverBase::Value &n)

   {

     LpSolverBase::Constr tmp(c);

     ///\todo Create an own exception type.

     if(!LpSolverBase::isNaN(tmp.upperBound())) throw LogicError();

     else tmp.upperBound()=n;

     return tmp;

   }

   ///\e

   ///\relates LpSolverBase::Constr

   ///

   inline LpSolverBase::Constr operator>=(const LpSolverBase::Value &n,

 					 const LpSolverBase::Constr&c) 

   {

     LpSolverBase::Constr tmp(c);

     ///\todo Create an own exception type.

     if(!LpSolverBase::isNaN(tmp.upperBound())) throw LogicError();

     else tmp.upperBound()=n;

     return tmp;

   }

   ///\e

   ///\relates LpSolverBase::Constr

   ///

   inline LpSolverBase::Constr operator>=(const LpSolverBase::Constr& c,

 					 const LpSolverBase::Value &n)

   {

     LpSolverBase::Constr tmp(c);

     ///\todo Create an own exception type.

     if(!LpSolverBase::isNaN(tmp.lowerBound())) throw LogicError();

     else tmp.lowerBound()=n;

     return tmp;

   }

   ///\e

   ///\relates LpSolverBase::DualExpr

   ///

   inline LpSolverBase::DualExpr operator+(const LpSolverBase::DualExpr &a,

                                           const LpSolverBase::DualExpr &b) 

   {

     LpSolverBase::DualExpr tmp(a);

     tmp+=b;

     return tmp;

   }

   ///\e

   ///\relates LpSolverBase::DualExpr

   ///

   inline LpSolverBase::DualExpr operator-(const LpSolverBase::DualExpr &a,

                                           const LpSolverBase::DualExpr &b) 

   {

     LpSolverBase::DualExpr tmp(a);

     tmp-=b;

     return tmp;

   }

   ///\e

   ///\relates LpSolverBase::DualExpr

   ///

   inline LpSolverBase::DualExpr operator*(const LpSolverBase::DualExpr &a,

                                           const LpSolverBase::Value &b) 

   {

     LpSolverBase::DualExpr tmp(a);

     tmp*=b;

     return tmp;

   }

   ///\e

   ///\relates LpSolverBase::DualExpr

   ///

   inline LpSolverBase::DualExpr operator*(const LpSolverBase::Value &a,

                                           const LpSolverBase::DualExpr &b) 

   {

     LpSolverBase::DualExpr tmp(b);

     tmp*=a;

     return tmp;

   }

   ///\e

   ///\relates LpSolverBase::DualExpr

   ///

   inline LpSolverBase::DualExpr operator/(const LpSolverBase::DualExpr &a,

                                           const LpSolverBase::Value &b) 

   {

     LpSolverBase::DualExpr tmp(a);

     tmp/=b;

     return tmp;

   }

 } //namespace lemon

 #endif //LEMON_LP_BASE_H