INFO

Contents:

1.   General Information
2.   General file format
2.1. Format (sections) of '.ieq'-files
2.2. Format (sections) of '.poi'- files
3.   Program calls and parameters


1. General Information:
-----------------------

PORTA is a collection of routines for analyzing polytopes and
polyhedra. The polyhedra are either given as the convex hull
of a set of points plus (possibly) the convex cone of a set of
vectors, specified in a '.poi'-file, or as a system of linear
equations and inequalities, specified in a '.ieq'-file - see
file format descriptions below. The following routines are
available:

dim      - computes the dimension of convex hull and convex
           cone for a set of given points by using a gaussian
           elimination algorithm. It displays the computed
           dimension as a terminal message and also writes to
           the end of the input file. If the input system is
           not full dimensional, the equations satisfied by
           the system are displayed.

fctp     - checks whether a set of inequalities given in a
           '.ieq'-file is facet inducing for a polyhedron given
           by a '.poi'-file. If this is not the case, points
           and rays which are not valid are output into a file.
           Points and rays that satisfy the inequality with
           equality are also determined and output into a file
           if there are any.

fmel     - reads a system of linear inequalities and eliminates
           choosen variables, i.e. given an index set I for
           variables to be eliminated, 'fmel' projects the given
           system to the subspace given by x_{i} = 0 for i in I.

iespo    - enumerates the subset of equations and inequalities
           in an 'ieq' input file which are valid (but not neces-
           sarily facet inducing) for a polyhedron given by a
           'poi' input file.

posie    - determines the number of points and direction vectors
           from a set given in a '.poi'-file which are valid for
           a system of equations and inqualities contained in a
           '.ieq.'-file.

portsort - puts the points or inequalities given in an input
           file into an increasing order according to the fol-
           lowing criteria:
             - right hand sides of inequalities or equations
             - frequency of the values -5 .. -1, 1 .. 5
             - lexicographical order
           Additionally 'portsort' formats the output.

traf     - carries out the transformation between the two poly-
           hedron representations, the direction is determined
           by the input filename suffix '.poi' or '.ieq' (see
           file format description below). All computations are
           carried out in rational arithmetic using integer ope-
           rations only to have guaranteed numerical results. A
           possible arithmetic overflow is recognized.

           The computation of  the  ieq-representation is perfor-
           med using Gaussian  and Fourier-Motzkin elimination.
           In the output file the right hand sides are 0, or de-
           termined by the smallest integer value for which the
           coefficients of the inequality are integral. If this
           is not possible with system integer arithmetic or if
           multiple precision  integer arithmetic is set, the
           right hand sides are 0 or 1 or -1 and the values are
           reduced as far as possible. If PORTA terminates
           successfully then the resulting inequalities
           are all facet-defining for your polyhedron and give
           together with equations a minimal linear description
           of your polyhedron.

           If an ieq-representation is given as input and if 0
           is not valid for the linear system, 'traf' needs a
           valid point that must be specified additionally in
           the input by using the keyword VALID - see format de-
           scription below. 'traf' transforms the ieq-represen-
           tation to the poi-representation, after elimination
           of equations and 0-centering, by applying the 'poi'-
           to-'ieq' direction to the polar polyhedron.
           Hint: If you give a valid point or if 0 is valid,
           then this vector may appear again in the resulting
           system, even if this vector might be redundant in
           a minimal description. (All other vectors are
           non-redundant.)

vint     - enumerates all integral points within given bounds
           that are valid for a linear system of inequalities
           and equations. The lower and upper bounds for each
           component must be specified in the input file by the
           keywords LOWER_BOUNDS and UPPER_BOUNDS - see format
           desription below.


2. General file format:
-----------------------

Files with name suffix '.ieq' contain a representation of a
polyhedron as a system of linear equations and inequalities.
Files with name suffix '.poi' contain a representation of a
polyhedron as the convex hull of a set of points and possibly
the convex cone of a set of vectors. The format is uniform
for both of '.ieq'- and '.poi'-files in having several sections
headed  by an indicator line with a specific capitalized key-
word, the first line stating the dimension <n> as

   DIM = <n>

and the last line containing the keyword

   END  .

The sections are specific to the '.ieq' and '.poi' polyhedron
representations with the exception of comment sections indica-
ted by the keyword

   COMMENT  ,

and a 'valid'-section which may appear in both types of files.

A 'valid'-section is headed by the keyword

   VALID

which indicates that the next line specifies a valid point for
the system of inequalities and equations by <n> rational values
in the format

   <numerator>/<denominator> ...

A denominator with value 1 can be omitted. A valid point is
required by the function 'traf' in case 0 is not valid for the
system.

There is no restriction concerning the order of sections and
some sections are optional. There are sections specific to PORTA
functions, such must be present in an input file for executing
the corresponding function.


2.1. Format (sections) of '.ieq'-files:
---------------------------------------

INEQUALITIES_SECTION

   Subsequent lines contain inequalities or equations, one per
   line, with format

   (<line>) <lhs> <rel> <rhs>

      <line> - line number (optional)

      <lhs>: <term{1}> +|- <term{2}>  ... +|- <term{n}>

         <term{i}>: <num{i}>/<den{i}> x{i} , i in {1,...<n>}

      <rel>: <= | >= | => | =< | = | ==

      <rhs>: <num_rhs>/<den_rhs>

   The values are rational, represented by numerators <num{i}>
   and denominators <den{i}>, i taken from {1,...,<n>}. A deno-
   minator with value 1 can be omitted.


LOWER BOUNDS

   The next line specifies lower bounds for the components of
   the system by <n> integer values such that the i-th entry re-
   fers to the i-th component. The lower bounds are used by the
   function 'vint' for enumerating integral points.


UPPER BOUNDS

   The next line specifies upper bounds for the components of
   the system by <n> integer values such that the i-th entry re-
   fers to the i-th component. The upper bounds are used by the
   function 'vint' for enumerating integral points.


ElIMINATION ORDER

   The next line specifies a set of variables to be eliminated
   by the function 'fmel' and the order of elimination by <n> in-
   teger values. A value 0 as the i-th entry of the line indicates
   that the i-th variable must not be eliminated, a value j, 0 < j,
   j < <n>, as the i-th entry of the line indicates that the i-th
   variable should be eliminated in the j-th iteration. All non-
   zero numbers must be different and it must be possible to put
   into an order 1,2,3,... .

See file 'example.ieq' for '.ieq' format illustration.


2.2. Format (sections) of '.poi'- files:
----------------------------------------

CONV_SECTION

   Subsequent lines contain specifications of points, one per
   line by <n> rational values, the line format is

   (<line>) <num{1}>/<den{1}> ...  <num{n}>/<num{n}>

      <line> - line number (optional)

      <num{i}> - i-th numerator

      <den{i}> - i-th denominator, a denominator with value 1
                 can be omitted

   If a CONV_SECTION is missing (the case of a cone) the origin
   is assumed to be feasible.


CONE_SECTION

   Subsequent lines contains specification of vectors, one per
   line, the line format is the same as for points.

See file 'example.poi' for '.poi' format illustration.

3. Program calls and parameters:
----------------------------

   dim [-pl] <filename>.poi

       p - Unbuffered redirection of terminal messages into
	   <filename>.prt

       l - Use a special integer arithmetic
           allowing the integers to have arbitrary lengths.
           This arithmetic is not as efficient as the system's
           integer arithmetic with respect to time and storage
           requirements.
           Note: Output values which exceed the 32-bit integer storage size
           are written in hexadecimal format (hex). Such hexadecimal
           format can not be reread as input.

   fctp <filename1>.ieq <filename2>.poi

       Filenames of output files are generated from <filename1>
       by appending the number of an corresponding inequality
       first and then the suffix '.poi' resp. '.poi.poi'.


   fmel [-pcl] <filename>.ieq

       p - Unbuffered redirection of terminal messages into
	   <filename>.prt

       c - Generation of new inequalities without the rule of
	   Chernikov.

       l - Use a special integer arithmetic
           allowing the integers to have arbitrary lengths.
           This arithmetic is not as efficient as the system's
           integer arithmetic with respect to time and storage
           requirements.
           Note: Output values which exceed the 32-bit integer storage size
           are written in hexadecimal format (hex). Such hexadecimal
           format can not be reread as input.

   iespo [-v] <filename1>.ieq <filename2>.poi

       v - Table indicating strong validity printed in the output
           file

       The output is written into a file the name of which is de-
       rived from <filename2> with suffix '.ieq'.


   posie <filename1>.ieq <filename2>.poi

       The output is written into a file with suffix '.poi' the
       name of which is derived from <filename1>.


   traf [-poscvl] <filename>.ieq    or
   traf [-poscvl] <filename>.poi

       p - Unbuffered redirection of terminal messages into
	   <filename>.prt

       o - Using a heuristic to eliminate that variable next,
	   for which the number of new inequalities is minimal
	   (local criterion). Inequalities which are recognized
	   to be facet-inducing for the finite linear system are
	   printed into a file as soon as they are identified.

       c - Fourier-Motzkin elimination without using the rule of
           Chernikov

       s - Statistical part appended to each line with the number
	   of coefficients

       v - Table indicating strong validity printed in the output
           file.

       l - Use a special integer arithmetic
           allowing the integers to have arbitrary lengths.
           This arithmetic is not as efficient as the system's
           integer arithmetic with respect to time and storage
           requirements.
           Note: Output values which exceed the 32-bit integer storage size
           are written in hexadecimal format (hex). Such hexadecimal
           format can not be reread as input.

   vint <filename>.ieq

       Output is written into a file named <filename>.poi.