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# Copyright (c) 1997-2024
# Ewgenij Gawrilow, Michael Joswig, and the polymake team
# Technische Universität Berlin, Germany
# https://polymake.org
#
# This program is free software; you can redistribute it and/or modify it
# under the terms of the GNU General Public License as published by the
# Free Software Foundation; either version 2, or (at your option) any
# later version: http://www.gnu.org/licenses/gpl.txt.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#-----------------------------------------------------------------------------
#
use application 'polytope';
# Determines the minimal entry of a vector.
sub vecMin {
my $vec=$_[0];
my $min=$vec->[0];
foreach(@$vec){
if($_<$min){
$min=$_;
}
}
return $min;
}
# Removes the (automatically added) trivial inequality
# 1,0,...,0 from a given set of inequalities.
sub removeTrivIneq {
my $ineqs_in=$_[0];
my $ineqs_wo_triv=$ineqs_in;
if ($ineqs_in->cols!=0) {
my $triv_ineq_row=-1;
my $triv_ineq=unit_vector<Rational>($ineqs_in->cols, 0);
for (my $i=0;$i<$ineqs_in->rows;$i++) {
if ($ineqs_in->row($i)==$triv_ineq) {
$triv_ineq_row=$i;
last;
}
}
if ($triv_ineq_row!=-1) {
$ineqs_wo_triv=new Matrix<Rational>($ineqs_in->minor(~[ $triv_ineq_row ], All));
}
}
return $ineqs_wo_triv;
}
# Converts a (rational) matrix (of inequalities) to an integer matrix
# and translates the inequality system into a system of equations
# by adding slack variables.
sub ineqs2IntEqsWithSlacks {
my ($ineqs_in) = @_;
my $ineqs=convert_to<Int>(primitive($ineqs_in));
my $slacks=unit_matrix<Int>($ineqs->rows);
my $eqs=$ineqs | -$slacks;
return $eqs;
}
# Checks whether a matrix contains only nonnegative entries.
# @return Bool true, if all entries are greater or equal than zero.
sub checkNonNeg {
my ($eqs)=@_;
my $answer=1;
for(my $i=0;$i<$eqs->rows;$i++){
for(my $j=0;$j<$eqs->cols;$j++){
if($eqs->[$i]->[$j]<0){
$answer=0;
return $answer;
}
}
}
return $answer;
}
# Generates the (Conti-Traverso) ideal <f_i-x_i>
# (or <f_i-x_i,w_1*...*w_m*t> for matrices with negative entries)
# and the monomial for the right-hand side
# from a given (integer) system of equations.
sub eqSys2ContiTrav {
my ($eqs_in)=@_; #eqs must be an integer matrix!
my $eqs=convert_to<Int>($eqs_in);
if($eqs_in!=$eqs){
die "Input matrix was not integer!";
}
$eqs->col(0)=-$eqs->col(0);# the first col is the right hand side of an eq system and must be multiplied by -1
my $isNonNeg=checkNonNeg($eqs);
my @vars=map { "w$_" } (1..$eqs->rows);
if(!$isNonNeg){
@vars = ( @vars, "t");
}
@vars=( @vars , map { "x$_" } (1..$eqs->cols-1) );
my @ideal;
my $isRHS=1;# the first col is the right hand side of eqs
my $bPoly;
for(my $j=0;$j<$eqs->cols;$j++){
my $monomVec=$eqs->col($j);
if (!$isNonNeg) {
$monomVec=$monomVec | 0;
my $minEntry=vecMin($monomVec);
if ($minEntry<0) {
$monomVec=$monomVec-$minEntry*ones_vector<Int>($eqs->rows+1);
}
}
$monomVec=$monomVec | zero_vector<Int>($eqs->cols);
if ($isRHS) {
$bPoly=new Polynomial(vector2row($monomVec), ones_vector<Rational>(1));
$isRHS=0;
} else {
my $x_i_pos=$eqs->rows+(1-$isNonNeg)+($j-1);
my $x_i_mono=unit_vector<Int>($monomVec->dim,$x_i_pos);
my $polyMatrix=$monomVec / $x_i_mono;
my $coeffs=new Vector<Rational>([1, -1]);
my $poly=new Polynomial($polyMatrix,$coeffs);
push(@ideal,$poly);
}
}
if (!$isNonNeg) {
my $negMonoVec=ones_vector<Int>($eqs->rows+1) | zero_vector<Int>($eqs->cols);
my $negMono=new Polynomial(vector2row($negMonoVec), ones_vector<Rational>(1));
my $negPoly=$negMono-1;
push(@ideal, $negPoly);
}
return [\@ideal, $bPoly, $r, \@vars];
}
# Produces the Singular input file for the Conti-Traverso procedure
# from the ideal and the RHS monomial generated in 'eqSys2ContiTrav'.
# The given cost vector c gives rise to an appropriate monomial ordering.
sub generateContiTravSingularInput {
my ($varsRef,$xcols,$c,$idealRef,$bPoly,$filename)=@_;
my @ideal=@$idealRef;
my @vars=@$varsRef;
open(OUT, ">$filename")
or die "can't create outputfile $filename: $!";
print(OUT "ring R = 0, (". join(",",@vars) . "), (lp(" . (scalar(@vars)-$xcols) . "),Wp(". join(",",@$c[1..$c->dim-1]) . "));\n");
print(OUT "ideal I = ". join(",",@ideal) .";\n");
print(OUT "ideal G = groebner(I);\n");
print(OUT "print(\"groebner:\");\n");
print(OUT "print(G);\n");
print(OUT "print(\"reduced RHS:\");\n");
print(OUT "reduce(" . $bPoly . ",G);\n");
print(OUT "quit;\n");
close(OUT);
return [$idealRef,$bPoly,$varsRef];
}
# Main function.
# Prepares the given H-description of a polyhedron //H_poly// for the
# Conti-Traverso procedure and calls the subroutines
# 'eqSys2ContiTrav' and 'generateContiTravSingularInput'.
# @param Polytope H_poly a polyhedron given in terms of inequalities and equations
# @param Rational c the (nonnegative) cost vector
# @param String filename the filename for the Singular input file
# @return [$idealRef,$bPoly,$varsRef]
sub contiTraverso {
my ($H_poly,$c,$filename)=@_;
my $facets_pre=$H_poly->give("FACETS | INEQUALITIES");
my $facets=removeTrivIneq($facets_pre);
my $aff_hull=$H_poly->give("AFFINE_HULL | EQUATIONS");
my $N_cols=max($facets->cols,$aff_hull->cols); #If FACETS or AFFINE_HULL are empty their matrix has zero cols!
my $eqs_in=convert_to<Int>(primitive($aff_hull));
my $c_in=convert_to<Int>(primitive($c));
if($facets->rows!=0){ #translate ineqs into eqs and adapt size of aff_hull
my $slackedIneqs=ineqs2IntEqsWithSlacks($facets);
my $N_slacks=$slackedIneqs->cols-$N_cols;
$eqs_in|=new Matrix<Int>($aff_hull->rows,$N_slacks);
$eqs_in/=$slackedIneqs;
$c_in|=new Vector<Int>(zero_vector<Int>($N_slacks));
}
my ($idealRef,$bPoly,$r,$varsRef) = @{eqSys2ContiTrav($eqs_in)};
my $xcols=$eqs_in->cols-1;
return generateContiTravSingularInput($varsRef,$xcols,$c_in,$idealRef,$bPoly,$filename);
}
################################################################################################
# End
################################################################################################
1; #to avoid return error
################################################################################################
# End of Code
################################################################################################
# Local Variables:
# mode: perl
# cperl-indent-level: 3
# indent-tabs-mode:nil
# End:
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