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/* -*- mode: C++; indent-tabs-mode: nil; -*-
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*
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* This file is a part of LEMON, a generic C++ optimization library.
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*
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* Copyright (C) 2003-2010
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* Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
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* (Egervary Research Group on Combinatorial Optimization, EGRES).
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*
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* Permission to use, modify and distribute this software is granted
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* provided that this copyright notice appears in all copies. For
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* precise terms see the accompanying LICENSE file.
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*
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* This software is provided "AS IS" with no warranty of any kind,
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* express or implied, and with no claim as to its suitability for any
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* purpose.
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*
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*/
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namespace lemon {
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/**
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[PAGE]sec_lp[PAGE] Linear Programming Interface
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Linear programming (LP) is one of the most important general methods of
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operations research. Countless optimization problems can be formulated
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and solved using LP techniques.
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Therefore, developing efficient LP solvers has been of high practical
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interest for a long time.
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Nowadays various efficient LP solvers are available, including both
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open source and commercial software packages.
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Therefore, LEMON does not implement its own solver, but it features
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wrapper classes for several known LP packages providing a common
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high-level interface for all of them.
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The advantage of this approach is twofold. First, our C++ interface is
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more comfortable than the typical native interfaces of the solvers.
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Second, changing the underlying solver in a certain application using
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LEMON's LP interface needs no effort. So, for example, one may try her
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idea using an open source solver, demonstrate its usability for a customer
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and if it works well, but the performance should be improved, then the
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customer may decide to purchase and use a better commercial solver.
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Currently, the following linear and mixed integer programming packages are
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supported: GLPK, Clp, Cbc, ILOG CPLEX and SoPlex.
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However, additional wrapper classes for new solvers can also be implemented
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quite easily.
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In this section, we will show two examples. The first one shows how simple
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it is to formalize and solve an LP problem in LEMON, while the second one
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shows how LEMON facilitates solving network optimization problems using LP
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solvers.
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\code
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Lp lp;
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Lp::Col x1 = lp.addCol();
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Lp::Col x2 = lp.addCol();
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lp.addRow(0 <= x1 + x2 <= 100);
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lp.addRow(2 * x1 <= x2 + 32);
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lp.colLowerBound(x1, 0);
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lp.colUpperBound(x2, 100);
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lp.max();
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lp.obj(10 * x1 + 6 * x2);
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lp.solve();
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std::cout << "Objective function value: " << lp.primal() << std::endl;
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std::cout << "x1 = " << lp.primal(x1) << std::endl;
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std::cout << "x2 = " << lp.primal(x2) << std::endl;
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\endcode
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\ref LpBase::Col "Lp::Col" type represents the variables in the LP problems,
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while \ref LpBase::Row "Lp::Row" represents the constraints. The numerical
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operators can be used to form expressions from columns and dual
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expressions from rows. Due to the suitable operator overloads,
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a problem can be described in C++ conveniently, directly as it is
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expressed in mathematics.
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The following example solves a maximum flow problem with linear
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programming. Several other graph optimization problems can also be
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expressed as linear programs and this interface helps to solve them easily
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(though usually not so efficiently as by a direct combinatorial method).
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\code
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Lp lp;
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ListDigraph::ArcMap<Lp::Col> f(g);
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lp.addColSet(f);
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// Capacity constraints
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for (ListDigraph::ArcIt a(g); a != INVALID; ++a) {
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lp.colLowerBound(f[a], 0);
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lp.colUpperBound(f[a], capacity[a]);
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}
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// Flow conservation constraints
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for (ListDigraph::NodeIt n(g); n != INVALID; ++n) {
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if (n == src || n == trg) continue;
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Lp::Expr e;
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for (ListDigraph::OutArcIt a(g,n); a != INVALID; ++a) e += f[a];
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for (ListDigraph::InArcIt a(g,n); a != INVALID; ++a) e -= f[a];
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lp.addRow(e == 0);
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}
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// Objective function
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Lp::Expr o;
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for (ListDigraph::OutArcIt a(g,src); a != INVALID; ++a) o += f[a];
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for (ListDigraph::InArcIt a(g,src); a != INVALID; ++a) o -= f[a];
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lp.max();
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lp.obj(o);
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lp.solve();
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std::cout << "Max flow value: " << lp.primal() << std::endl;
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\endcode
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[TRAILER]
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*/
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}
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