#### D.4.13.1 solve_IP

Procedure from library `intprog.lib` (see intprog_lib).

Usage:
  solve_IP(A,bx,c,alg); A intmat, bx intvec, c intvec, alg string.
  solve_IP(A,bx,c,alg); A intmat, bx list of intvec, c intvec, alg string.
  solve_IP(A,bx,c,alg,prsv); A intmat, bx intvec, c intvec, alg string, prsv intvec.
  solve_IP(A,bx,c,alg,prsv); A intmat, bx list of intvec, c intvec, alg string, prsv intvec.

Return:
  same type as bx: solution of the associated integer programming problem(s) as explained in

  Toric ideals and integer programming.

Note:
  This procedure returns the solution(s) of the given IP-problem(s) or the message `not solvable'.
  One may call the procedure with several different algorithms:
  - the algorithm of Conti/Traverso (ct),
  - the positive variant of the algorithm of Conti/Traverso (pct),
  - the algorithm of Conti/Traverso using elimination (ect),
  - the algorithm of Pottier (pt),
  - an algorithm of Bigatti/La Scala/Robbiano (blr),
  - the algorithm of Hosten/Sturmfels (hs),
  - the algorithm of DiBiase/Urbanke (du).
  The argument `alg' should be the abbreviation for an algorithm as above: ct, pct, ect, pt, blr, hs or du.

  `ct' allows computation of an optimal solution of the IP-problem directly from the right-hand vector b.
  The same is true for its `positive' variant `pct' which may only be applied if A and b have nonnegative entries.
  All other algorithms need initial solutions of the IP-problem.

  If `alg' is chosen to be `ct' or `pct', bx is read as the right hand vector b of the system Ax=b. b should then be an intvec of size m where m is the number of rows of A.
  Furthermore, bx and A should be nonnegative if `pct' is used. If `alg' is chosen to be `ect',`pt',`blr',`hs' or `du', bx is read as an initial solution x of the system Ax=b. bx should then be a nonnegative intvec of size n where n is the number of columns of A.

  If `alg' is chosen to be `blr' or `hs', the algorithm needs a vector with positive coefficients in the row space of A.
  If no row of A contains only positive entries, one has to use the versions of solve_IP which take such a vector prsv as an argument.

  solve_IP may also be called with a list bx of intvecs instead of a single intvec.

Example:

    LIB "intprog.lib";
    // 1. call with single right-hand vector
    intmat A[2][3]=1,1,0,0,1,1;
    intvec b1=1,1;
    intvec c=2,2,1;
    intvec solution_vector=solve_IP(A,b1,c,"pct");
    solution_vector;"";
    ==> 0,1,0
    ==>
    // 2. call with list of right-hand vectors
    intvec b2=-1,1;
    list l=b1,b2;
    l;
    ==> [1]:
    ==>    1,1
    ==> [2]:
    ==>    -1,1
    list solution_list=solve_IP(A,l,c,"ct");
    solution_list;"";
    ==> [1]:
    ==>    0,1,0
    ==> [2]:
    ==>    not solvable
    ==>
    // 3. call with single initial solution vector
    A=2,1,-1,-1,1,2;
    b1=3,4,5;
    solve_IP(A,b1,c,"du");"";
    ==> 0,7,2
    ==>
    // 4. call with single initial solution vector
    //    and algorithm needing a positive row space vector
    solution_vector=solve_IP(A,b1,c,"hs");"";
    ==> ERROR: The chosen algorithm needs a positive vector in the row space of t\
       he matrix.
    ==> 0
    ==>
    // 5. call with single initial solution vector
    //     and positive row space vector
    intvec prsv=1,2,1;
    solution_vector=solve_IP(A,b1,c,"hs",prsv);
    solution_vector;"";
    ==> 0,7,2
    ==>
    // 6. call with list of initial solution vectors
    //    and positive row space vector
    b2=7,8,0;
    l=b1,b2;
    l;
    ==> [1]:
    ==>    3,4,5
    ==> [2]:
    ==>    7,8,0
    solution_list=solve_IP(A,l,c,"blr",prsv);
    solution_list;
    ==> [1]:
    ==>    0,7,2
    ==> [2]:
    ==>    7,8,0

See also: Integer programming; intprog_lib; toric_lib.

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