COIN-OR::LEMON - Graph Library

Changeset 687:6c408d864fa1 in lemon for doc/groups.dox


Ignore:
Timestamp:
04/29/09 03:15:24 (10 years ago)
Author:
Peter Kovacs <kpeter@…>
Branch:
default
Phase:
public
Message:

Support negative costs and bounds in NetworkSimplex? (#270)

  • The interface is reworked to support negative costs and bounds.
    • ProblemType? and problemType() are renamed to SupplyType? and supplyType(), see also #234.
    • ProblemType? type is introduced similarly to the LP interface.
    • 'bool run()' is replaced by 'ProblemType? run()' to handle unbounded problem instances, as well.
    • Add INF public member constant similarly to the LP interface.
  • Remove capacityMap() and boundMaps(), see also #266.
  • Update the problem definition in the MCF module.
  • Remove the usage of Circulation (and adaptors) for checking feasibility. Check feasibility by examining the artifical arcs instead (after solving the problem).
  • Additional check for unbounded negative cycles found during the algorithm (it is possible now, since negative costs are allowed).
  • Fix in the constructor (the value types needn't be integer any more), see also #254.
  • Improve and extend the doc.
  • Rework the test file and add test cases for negative costs and bounds.
File:
1 edited

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  • doc/groups.dox

    r658 r687  
    353353in a network with capacity constraints (lower and upper bounds)
    354354and arc costs.
    355 Formally, let \f$G=(V,A)\f$ be a digraph,
    356 \f$lower, upper: A\rightarrow\mathbf{Z}^+_0\f$ denote the lower and
     355Formally, let \f$G=(V,A)\f$ be a digraph, \f$lower: A\rightarrow\mathbf{Z}\f$,
     356\f$upper: A\rightarrow\mathbf{Z}\cup\{+\infty\}\f$ denote the lower and
    357357upper bounds for the flow values on the arcs, for which
    358 \f$0 \leq lower(uv) \leq upper(uv)\f$ holds for all \f$uv\in A\f$.
    359 \f$cost: A\rightarrow\mathbf{Z}^+_0\f$ denotes the cost per unit flow
    360 on the arcs, and \f$sup: V\rightarrow\mathbf{Z}\f$ denotes the
     358\f$lower(uv) \leq upper(uv)\f$ must hold for all \f$uv\in A\f$,
     359\f$cost: A\rightarrow\mathbf{Z}\f$ denotes the cost per unit flow
     360on the arcs and \f$sup: V\rightarrow\mathbf{Z}\f$ denotes the
    361361signed supply values of the nodes.
    362362If \f$sup(u)>0\f$, then \f$u\f$ is a supply node with \f$sup(u)\f$
    363363supply, if \f$sup(u)<0\f$, then \f$u\f$ is a demand node with
    364364\f$-sup(u)\f$ demand.
    365 A minimum cost flow is an \f$f: A\rightarrow\mathbf{Z}^+_0\f$ solution
     365A minimum cost flow is an \f$f: A\rightarrow\mathbf{Z}\f$ solution
    366366of the following optimization problem.
    367367
     
    405405The dual solution of the minimum cost flow problem is represented by node
    406406potentials \f$\pi: V\rightarrow\mathbf{Z}\f$.
    407 An \f$f: A\rightarrow\mathbf{Z}^+_0\f$ feasible solution of the problem
     407An \f$f: A\rightarrow\mathbf{Z}\f$ feasible solution of the problem
    408408is optimal if and only if for some \f$\pi: V\rightarrow\mathbf{Z}\f$
    409409node potentials the following \e complementary \e slackness optimality
     
    414414   - if \f$lower(uv)<f(uv)<upper(uv)\f$, then \f$cost^\pi(uv)=0\f$;
    415415   - if \f$cost^\pi(uv)<0\f$, then \f$f(uv)=upper(uv)\f$.
    416  - For all \f$u\in V\f$:
     416 - For all \f$u\in V\f$ nodes:
    417417   - if \f$\sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \neq sup(u)\f$,
    418418     then \f$\pi(u)=0\f$.
    419419 
    420420Here \f$cost^\pi(uv)\f$ denotes the \e reduced \e cost of the arc
    421 \f$uv\in A\f$ with respect to the node potentials \f$\pi\f$, i.e.
     421\f$uv\in A\f$ with respect to the potential function \f$\pi\f$, i.e.
    422422\f[ cost^\pi(uv) = cost(uv) + \pi(u) - \pi(v).\f]
    423423
    424 All algorithms provide dual solution (node potentials) as well
     424All algorithms provide dual solution (node potentials) as well,
    425425if an optimal flow is found.
    426426
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