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## NAME

```       lrslib - Convert between represetations of convex polyhedra.

```

## SYNOPSIS

```       lrs input.ine

lrs input.ine | lrsbuffer

lrsfourier file.ine [fileout]

redund input.ine

```

## DESCRIPTION

```       A polyhedron can be described by a list of inequalities
(H-representation) or as by a list of its vertices and extreme rays
(V-representation).  lrs is a C program that converts a
H-representation of a polyhedron to its V-representation, and vice
versa.  These problems are known respectively at the vertex enumeration
and convex hull problems.

Fukuda's FAQ page[1]   contains a more detailed introduction to the
problem, along with many useful tips for the new user.

lrsbuffer can remove some duplicate output.  redund finds redundant
inequalities in the input.

```

## FILEFORMATS

```       File formats were developed jointly with Komei Fukuda and are
compatible with cdd[2].

The input for lrs is a H- or V- representation of a polytope.

name
{representation line}
{options}
{linearities[3]}
begin
m n rational
{input matrix}
end
{options}

name is a user supplied name for the polytope.  Comments may appear
before the begin or after the end, and to avoid interpretation as an
option, should begin with a special character such as "*" or "#".

name is a user supplied name for the polytope.  representation line is
either "H-representation" or "V-representation". If is omitted,
H-representation is assumed.  The input coefficients are read in free
format, and are not checked for type. Coefficients are separated by
white space. m is the number of rows and n the number of columns of the
input matrix.

H-representation
The integer  m is the number of inequalities,  and the integer n is the
dimension of the input +1. A list of inequalities contains the
coefficients of inequalities of the form

a0 + a1x1+ ... + an-1 xn-1 >=  0.

This inequality is input as the line

a0 a1... an-1

The coefficients can be entered as integers or rationals in the format
x/y.

V-representation
The integer  m is the number of vertices and rays,  and the integer n
is the dimension of the input +1. Each vertex is given in the form

1   v0   v 1...   vn-1

Each ray is given in the form

0   r0   r 1...   rn-1

where r0   r 1...   rn-1is a point on the ray.

There must be at least one vertex in each file. For bounded polyhedra
there will be no rays entered. The coefficients can be entered as
integers or rationals in the format x/y.

Note for cdd users: lrs uses essentially the same file format as cdd.
Files prepared for cdd should work with little or no modification. Note
that  the V-representation corresponds to the "hull" option in cdd.
Options specific to cdd can be left in the input files and will be
ignored by lrs.  Note the input files for lrs are read in free format,
after the line m n rational, lrs will look for exactly m*n rationals or
integers separated by white space (blank,  carriage return, tab etc.).
lrs will not "drop" extra columns of input if n is less than the number
of columns supplied.

Basic Options
Almost all options are placed after the end statement, maintaining
compatibility with cdd. Where this is not the case, it will be
mentioned explicitly.

allbases This option instructs lrs to list each vertex (or facet) for
each of its bases.  Output Duplication[4] .[5] This option is often
combined with printcobasis.

bound  x Use with H-representation  - for lrs or nash Either the
maximize or minimize option should be selected. x is an integer or
rational. For maximization (resp. minimization) the reverse search tree
is truncated  whenever the current objective value is less (resp. more)
than x.

cache n lrs stores the latest  n dictionaries in the reverse search
tree. This speeds up the backtracking step, but requires more memory.

debug  startingbasis endingbasis Print out cryptic but detailed trace,
dictionaries etc. starting at #B=startingbasis and ending at
#B=endingbasis. debug 0 0 gives a complete trace.

digits n
placed before the begin statement n is the maximum number of decimal
digits to be used. If this is exceeded the program terminates with a
message (it can  usually be restarted).   The default is set to about
100 digits. At the end of a run a message is given informing the user
of the maximum integer size encountered. This may be used to optimize
memory usage and speed on subsequent runs (if doing estimation for
example).

dualperturb If lrs is executed with the maximize or minimize option,
the reverse search tree is rooted at an optimum vertex for this
function.If there are mulitiple optimum vertices, the output will often
not be complete. This option gives a small perturbation to the
objective to avoid this. A warning message is given if the starting
dictionary is dual degenerate.

estimates k Estimate the output size. Used in conjunction with maxdepth
- see Estimation.[6]

geometric   // H-representation  or voronoi option only // With this
option, each ray is printed together with the vertex with which it is

incidence This option automatically switches on printcobasis , so see
below for a description of this option first. Can be used with
printcobasis n. (Ver 4.2b) .PP For input H-representation, indices of
all input inequalities that contain the vertex/ray that is about to be
output. For a simplicial face, there is no new output, since these
inequalities are listed after a colon. .PP For input V-representation,
indices of all input vertices/rays that lie on the facet that is about
to be output. A starred indexindicates that this vertex  is also in the
cobasis, but is not contained in the facet. It arises due to the
lifting operation used with input V-representations.

#incidence The same as printcobasis. Included for compatability with
cdd.

linearity  k  i1i2 i ... ik The input contains k linearities in rows
i1i2i ... ikof the input file are equations. See Linearities.[3]

maxdepth k The search will be truncated at depth k. All bases with
depth less than or equal to k will be computed.  k is  a non-negative
integer, and this option is used for estimates - see Estimation.[6]
Note: For H-representations, rays at depth k will not be reported. For
V-representations, facets at depth k will not be reported.

maximize a0 a1... an-1  // H-representation  only //

minimize  a0 a1... an-1 // H-representation  only //

If used with lrs the starting vertex maximizes (or minimizes) the
function  a0 + a1x1+ ... + an-1 xn-1.The dualperturb option may be
needed to avoid dual degeneracy.See Nash Equilibria and  Linear
Programming[7]

maxoutput n Limits number of output lines produced (either
vertices+rays or facets) to n

mindepth k  Backtracking will be terminated at depth k, for k a
non-negative integer. This can be used for running reverse search on
subtrees as separate processes, e.g. in a distributed computing
environment.

nonnegative // This option must come before the begin statement//
//H-representation only //   Bug: Can only be used if the origin is a
vertex of the polyhedron  For problems where the input is an
H-representation of the form b+Ax>=0, x>=0 (ie. all variables
non-negative, all constraints inequalities) it is not necessary to give
the non-negative constraints explicitly if the nonnegative option is
used. This option cannot be used for V-representations, or with the
linearity option (in which case the linearities will be treated as
inequalities). This option may be used with redund , but the implied
nonnegativity constraints are not tested themselves for redundancy. To
test everything it is necessary to enter the nonnegativity constraints
explicitly in the input file. (In Ver 4.1, the origin must be a
vertex).

printcobasis  k;Modified in lrs 4.0 Every k'th cobasis is printed. If k
is omitted, the cobasis is printed for each vertex/ray/facet that is
output. For a long run it is useful to print the cobasis occasionally
so that the program can be restarted if necessary.
H-representation: If the input is an H-representation the cobasis is a
list the indices of the inequalities from the input file that define
the current vertex or ray. See option  incidence above for more
information. For rays, a cobasis is also printed. In this case the
cobasis is the cobasis of the vertex from which the ray emanates. One
of the indices is starred, this indicates the inequality to be dropped
from the cobasis to define the ray. Alternatively, if the
allbasesoption is used, all cobases will be printed out.
V-representation: If the input is a V-representation, the cobasis is a
list of the input vertices /rays that define the current facet. See
facet all 4 indices must be given in this order (omit the *).

printslack New in Ver 4.2 ; // Use with H-representation // lrs prints
a list of the indices of the input inequalities that are satisfied
strictly for the current vertex, ie. corresponding slack variable is
positive. If nonnegative is set, the list will also include indices n+i
for each decision variable xi which is positive.  project Used by
lrsfourier[8] only.

restart  V# R# B# depth {facet #s or vertex/ray #s} Modified in lrs4.0
lrs can be restarted from any known cobasis. The calculation will
proceed to normal termination. All of the information is contained in
the output from a printcobasis option.  The order of the indices is
very important, enter them exactly as they appear in the output from
the previously aborted run.

startingcobasis i1i2i ... in-1 This allows the user to specify a known
cobasis for beginning the reverse search.  i1i2i ... in-1 is a list of
the inequalities (for H-representation) or vertices/rays (for
V-representation) that define a cobasis. If it is invalid, or this
option is not specified, lrs will find its own starting cobasis.  The
reverse search tree is truncated(pruned)  whenever a new vertex is
encountered. Note: This does note necessarily produce the set of all
vertices adjacent to the optimum vertex in the polyhedron, but just a
subset of them.

verbose Print slightly more detailed information about the run.

volume // V-representation  only // Compute volume - see section Volume
Computation.[9]

voronoi // V-representation  only - place immediately after end
statement // Compute Voronoi diagram - see section Voronoi
Diagrams.[10]

```

## NOTES

```        1. FAQ page
http://www.ifor.math.ethz.ch/staff/fukuda/polyfaq/polyfaq.html

2. cdd
http://www.cs.mcgill.ca/%7Efukuda/soft/cdd_home/cdd.html

3. linearities
http://cgm.cs.mcgill.ca/%7Eavis/C/lrslib/USERGUIDE.html#Linearities

4. Output Duplication
http://cgm.cs.mcgill.ca/%7Eavis/C/lrslib/USERGUIDE.html#Output%20Duplication

5.

6. Estimation.
http://cgm.cs.mcgill.ca/%7Eavis/C/lrslib/USERGUIDE.html#Estimation

7. Linear Programming
http://cgm.cs.mcgill.ca/%7Eavis/C/lrslib/USERGUIDE.html#Linear%20Programming

8. lrsfourier
http://cgm.cs.mcgill.ca/%7Eavis/C/lrslib/USERGUIDE.html#fourier

9. Volume Computation.
http://cgm.cs.mcgill.ca/%7Eavis/C/lrslib/USERGUIDE.html#Volume%20Computation

10. Voronoi Diagrams.
http://cgm.cs.mcgill.ca/%7Eavis/C/lrslib/USERGUIDE.html#Voronoi%20Diagrams
```