User's Guide for lrs - Version 4.2

June 1, 2005                    Copyright(C) 2005                  lrs home page

  David Avis          avis@cs.mcgill.ca     http://cgm.cs.mcgill.ca/~avis

• nash:  Computes all Nash equlibria of a two person non-cooperative game. Can handle bounds on payoff functions.
  
• fourier: Projects a polyhedron given by an H-representation onto a subspace using Fourier elimination (contributed by Tallman Nkgau).
  

What's New in Version 4.1

A new feature of the basic package is the nonnegative option that speeds up the solution of problems with H-representation:

b+Ax >= 0,        x>=0

What's New in Version 4.0 (included in Version 4.1)

This distribution also comes with three  arithmetic lpackages: lrsmp, lrslong and lrsgmp . The library lrsmp is essentially the same extended precision package included in earlier distributions. lrslong is an implementation using long integer arithmetic, which is considerably faster, but does no overflow checking. lrsgmp is an interface to GNU MP which must be installed first. Binaries are produced for all arithmetic packages using the distributed make file.

Most of the functions required for lrs and redund are included in the library lrslib. This is compiled with either of the arithmetic packages. One of the purposes of supplying a library is to allow simple customization of the reverse search procedure. The program lrs.c is the reverse search driver, and supplies each output vector to the user. It can be customized to prune the search according to various criteria, eg., if the value of some objective function falls below some value, or if an integer vector is produced etc.

The installation procedure has been simplified, and the package is distributed as a compressed tar file. Included is a makefile for various configurations and some typical input files. Binaries are also included for some platforms.

There are new options:  incidence , which lists all inequalities (resp. vertices/rays) which are incident with the current output vertex/ray (resp. facet). and truncate which prunes the search tree everytime a vertex different from the starting vertex is reached.

Introduction

lrs is based on the reverse search algorithm of Avis and Fukuda(1992) , modified to use lexicographic pivoting  and implemented in rational arithmetic. See Avis(1998a) for a technical description, and Avis(1998b) for some computational experience. The input files are in Polyhedra format , developed by Fukuda and the author. The format is essentially self-dual, and the output file produced can be read in as an input file, with very minor modifications, to perform the reverse transformation. This format is compatible with that  used in Fukuda's cdd package, which performs the same transformations using a version of the double description method.  cdd can also be used  in conjunction with lrs as a pre-processor for projections to subspaces, or as a post processor for computing the entire face lattice. Another program using the same file format is the primal-dual method pd, developed by Bremner, Fukuda and Marzetta .  It is essentially dual to lrs, and is very efficient for computing H-representations of simple polyhedra, and V-representations of simplicial polyhedra. It will compute the volume of a polytope given by an H-representation.

Another program  based on the double description method is Christof and Loebel's porta , and a versatile tool for the algorithmic treatment of polytopes is Gawrilow and Joswig's  polymake package. Barber et al.'s qhull.  A package for volume computation called Vinci has been developped by Enge. A comprehensive general source of related infomation are Erickson's Computational Geometry Pages.

In version 4.0, polyhedra handled by lrs  need not be full dimensional  and may contain input linearities and redundant columns .  lrs accepts either integer or rational input, and produces integer or rational output. All computations are done exactly using either extended precision arithmetic or fixed long integer arithmetic. In the latter case no overflow checking is performed, and the user is advised to check results using the extended precision version. Since it is a pivot based method, lrs can be very slow for degenerate inputs: i.e.. H-representations of non-simple polyhedra, and V-representations of non-simplicial polyhedra. On the other hand, it does not store the vertices/ rays or facets produced, so for very large problems it may be the only method that can solve the problem.  A discussion of various vertex enumeration/convex hull methods and the types of polyhedra that cause them to behave badly is contained in Avis, Bremner and Seidel( 1997).

Additional functions of lrs include:

• Estimating the number of vertices/rays or facets of a polytope, and estimating the total running time
• Computing the volume of a polytope given by a V-representation
• Solving LP problems for a polyhedron given by a H-representation
• Computing the Voronoi vertices and rays for an input set of data points
• The ability to suspend and restart execution at any time

These programs can be distributed freely under the GNU GENERAL PUBLIC LICENSE. Please read the file COPYING carefully before using.  Please inform the author of any interesting applications for which lrs/redund were helpful.

---

Installation

• Some binaries are available. From lrs home page,  click on  "Download" and change directory to "binaries" and check for available platforms. These should be usable directly, however to get source, full documentation and a set of test files, you will need to retrieve lrslib-042 as described below.

 
• From lrs home page, click on "Download" and retrieve the file lrslib-042.tar.gz

 
• Unpack with:

   % gunzip lrslib-042.tar.gz
   % tar xvf lrslib-042.tar
 
• Go to the new directory

% cd lrslib-042
 
•  make all binariesl (delete those not needed afterwards)
  

   % make all               (most 32 bit unix machines)
or
  % make all64              (64 bit integer machines such as DEC Alpha)
or
  % make nosigs        ( ansi standard version for 32 bit machines without signals/timing routines)
 
• If you have GNU MP installed you can obtain binaries glrs and gredund using this library by typing

  % make gmp
It may be necessary to edit makefile to set the paths to the gmp library
 
 
• Test the program lrs by typing:
lrs  cube.ine
This will convert the H-representation of a cube given cube by 6 the six inequalities -1 <= x_i <= 1 , i=1,2,3 given by the input:
cube
*cube of side 2 centred at the origin
H-representation
begin
6 4 rational
1 1 0 0
1 0 1 0
1 0 0 1
1 -1 0 0
1 0 -1 0
1 0 0 -1
end

The output should contain (along with other comment lines beginning with a "*")

cube
V-representation
begin
***** 4 rational
 1  1  1  1
 1 -1  1  1
 1  1 -1  1
 1 -1 -1  1
 1  1  1 -1
 1 -1  1 -1
 1  1 -1 -1
 1 -1 -1 -1
end

This is a list of the 8 vertices with each co-ordinate +/- 1.  The ***** should be replaced by the actual number, 8, of vertices. Since lrs does not save the output produced, it does not know this value until the execution terminates. This output is now essentially the same as file cube.ext. To complete the test type:
 

lrs  cube.ext
Now the output produced is essentially the file cube.ine, with the inequalities appearing in a different order.

The program buffer is a small utility that can be used to remove duplicate line in the output that may occur if the input is not a polytope or a pointed cone. For such unbounded polyhedra,  rays may appear in the output more than once. The program takes two optional integer parameters:

buffer  maxline maxbuffer
It builds a circular buffer of maxbuffer (default 50) lines of maxline (default 5000)characters each. If an output line is already in the buffer, it is not output.  To use buffer, pipe the output from lrs:
lrs  filename | buffer

---

File Formats

Example files can be found in the directories ine and ext which come in the standard distribution. File formats were developed jointly with Komei Fukuda and are compatible with cdd . The cdd manual contains additional information and examples.

The input for lrs is a H- or V- representation of a polytope. These files have the following formats:

H-representation

---


name
H-representation
{options}
{linearities }                                /* new in version 4.0 */
begin
m n rational

{list of inequalities }

end
{options}

---


 

name is a user supplied name for the polytope.  If the line H-representation 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. Normally this file would be saved with filename suffix .ine but this not required.  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 "#".
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 + a₁ x₁ + ... + a_n-1 x_n-1 >=  0.

This inequality is input as the line

a₀ a₁ ... a_n-1
 

The coefficients can be entered as integers or rationals in the format x/y.
For example, the square centred at the origin with side length two has inequalities

1 + x₁ >= 0      1+ x₂ >=0         1-x ₁ >= 0           1-x₂ >=0

and would be represented by the input file

square
*centred square of side 2
H-representation
begin
4 3 rational
1 1 0
1 0 1
1 -1 0
1 0 -1
end
 

V-representation

---


name
V-representation
{options}
{linearities }                             /* new in version 4.0 */
begin
m n rational

{list of vertices and extreme rays}
end
{options)


---


name is a user supplied name for the polytope.  The line V-representation is required.  The input coefficients are read in free format, and are not checked for type. Coefficients are separated by white space. 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 "#".Normally this file would be stored with filename suffix . ext, but this is not required.

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 ...   v_n-1
 

Each ray is given in the form

0   r₀   r ₁ ...   r_n-1

where r₀   r ₁ ...   r _n-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.
For example, the unit square centred at the origin has vertices
(1,1) ,(1,-1),(-1,1),(-1,-1)
and would be represented by the input file

square
*centred square of side 2
V-representation
begin
4 3 rational
1 1 1
1 1 -1
1 -1 1
1 -1 -1
end

The positive quadrant has vertex (0,0) and rays (1,0) (0,1) and is represented

quadrant
*positive quadrant
V-representation
begin
3 3 rational
1 0 0
0 1 0
0 0 1
end

Its H-representation contains the inequalities x₁ >= 0 and x₂ >= 0 :

quadrant
*positive quadrant
H-representation
begin
2 3 rational
0 1 0
0 0 1
end
 
 

Basic Options

lrs has many options to allow various functions to be performed, and to modify the output produced. Almost all options are placed after the end statement, maintaining compatibility with cdd. Where this is not the case, it will be mentioned explicitly. All options are optional.

This option instructs lrs to list each vertex (or facet) for each of its bases. Normally a vertex (or facet) is only output when its lex-min basis is found, to avoid duplications - see section Output Duplication . This option is often combined with printcobasis .

lrs stores the latest  n dictionaries in the reverse search tree. This speeds up the backtracking step, but requires more memory.  If n is set to one, there is no caching and "pure" reverse search is performed.  The default is n=10. At the end of a run a message gives cache information. The output
*Dictionary Cache: max size= 4 misses= 10/1340      Tree Depth=5
indicates that with cache size set to 4, only 10  of the 1340 dictionaries computed were not in the cache  during backtracking and had to be recomputed. The maximum tree depth was 5, so there would be no misses with a cache size of 6. Full caching reduces processing time by about 40%.

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

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). The output:
*Max digits= 45/100
indicates that the maximum integer encountered had 45 decimal digits, and the program allowed up to 100 digit integers.

Estimate the output size. Used in conjunction with maxdepth - see Estimation.

With this option, each ray is printed together with the vertex with which it is incident. They output has the form:

   0   r₀   r ₁ ...   r_n-1* 1   v₀   v ₁ ...   v_n-1

1 0 0
0 0 1 * 1 0 0
0 1 0 * 1 0 0
This indicates the origin is a vertex, and there are two geometric rays: (0,t) and (t,0) both adjacent to the origin. For more information see Geometric Rays in Hints and Comments .

This option automatically switches on printcobasis , so see below for a description of this option first.

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 indices are already listed. Otherwise, the additional tight inequalities are listed after a colon. Eg:
V#1 R#0 B#1 h=0 facets  12 14 15 16 : 9 10 11 13 I#8 det= 8
 1  0  0  0  1
The vertex 0 0 0 1 satisfies 8 input inequalities as equations, as indicated by I#8 : those with indices 12,14,15,16 are in the cobasis, and those with indices 9, 10, 11, 13 are in the basis. For a ray:
V#1 R#5 B#1 h=0 facets  5 9* 10 11 12 13 : 2 3 4 I#8 det= 8
 0  1  1  0  0  1  1
Here the ray 1  1  0  0  1  1 lies on 8 inequalities, with indices 5 10 11 12 13 in basis and 2 3 4 in cobasis. The starred index 9* indicates that the ray is terminated by the input inequality 9. This inequality is in the cobasis and defines the vertex from which the ray starts.

For input V-representation, indices of all input vertices/rays that lie on the facet that is about to be output:
F#5 B#3 h=2 vertices/rays  7 8* 11 13 15 : 1 3 5 9 I#8 det= 16
1 -1  0  0  0
The facet generated by inequality x₁ <= 1 contains 8 input vertices, as indicated by I#8: those with indices 7,11,13,15 are in the cobasis, and those with indices 1 3 5 9 are in the basis.The starred index 8* indicates 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.

The same as printcobasis. Included for compatability with cdd.

The input contains k linearities in rows i₁ i₂ i ... i_k of the input file are equations. See Linearities.

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.
Note: For H-representations, rays at depth k will not be reported. For V-representations, facets at depth k will not be reported.

 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.

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 lrs4.0

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. For example, with input cube.ine a typical output is:
V#5 R#0 B#5 h=1 facets  3 4 5 I#3 det=1
1 1 1 -1
This indicates that the vertex (1,1,-1) is defined by the 3rd, 4th and 5th facet inequalities in the input file being satisfied as equations. It is the (B#)5th basis computed, and there have been (V#)5 vertices and (R#)0 rays output up to this point. I#3 means that this vertex is incident with 3 input inequalities, ie. it is non-degenerate. 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. For example the output:
V#1 R#6 B#4 h=1 facets  2 4* 5 7 I#3 det=1
0 1 1 2 1
indicates that the ray (1,1,2,1) emanates from a vertex with cobasis defined by input inequalities 2 4 5 7 satisfied as equations. The ray is defined by dropping the equation with index 4. Note that there may not appear any vertex in the output with cobasis 2 4 5 7, since the corresponding vertex may be degenerate and printed with another (the lex-min) cobasis. To find out which vertices correspond to which rays, use also the geometric option.  Now the output may appear:
V#1 R#6 B#4 h=1 facets  2 4* 5 7 I#3 det=1
0 1 1 2 1  * 1 0 0 0 0
This indicates that the ray is incident to the origin. Alternatively, if the allbases option 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. For example, with input file cube.ext a typical output is:
F#5 B#4 h=3 vertices/rays  2 3 4 5* I#3 det= 8
 1  0  0 -1
There have been 5 facets output up to this point, and 4 bases have been computed. This facet is defined by the vertices in positions 2,3,4 in the input file. The additional cobasis index 5* appears because a V-representation is lifted to one higher dimension before processing, and this index fills out the cobasis. I#3 means that this facet is incident with 3 input vertices/rays, ie. it is non-degenerate. See option incidence above for more information.
To initiate lrs from this facet all 4 indices must be given in this order (omit the *), eg.
startingcobasis 2 3 4 5
Similarly, all 4 indices must be given in order to restart from this facet:
restart 5 4 3 2 3 4 5

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. To restart from cobasis:

V#5 R#0 B#5 h=1 facets  3 4 5 det=1
enter:
restart 5 0 5 1 3 4 5

F#5 B#4 h=3 vertices/rays  2 3 4 5* det= 8
 1  0  0 -1
enter
restart 5 4 3 2 3 4 5

Note that if some cobasic index is followed by a "*",  then the index only, without the "*", is included in the restart line.
Caution: When restarting, output from the restart dictionary may be duplicated, and the final totals of number of vertices/rays/facets may reflect this.

This allows the user to specify a known cobasis for beginning the reverse search. i₁ i₂ i ... i_n-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. For example, with cube.ine, the user can start at vertex (-1,1,-1) by specifying:
startingcobasis 1 3 4

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.

Print slightly more detailed information about the run.

Compute volume - see section Volume Computation.

Compute Voronoi diagram - see section Voronoi Diagrams.

Arithmetic Packages                      (new in Version 4.0)

   lrsmp     Extended precision arithmetic used in previous releases

   lrslong   Fixed length long integer arithmetic. No overflow checking
                    but 5-10 times faster than lrsmp.

   lrsgmp   An interface to GNU MP  which must be installed first.
                    Runs 1-6 times faster than lrsmp on typical problems.

  The standard make gives binaries  lrs/redund with lrsmp, and lrs1/redund1 with lrslong. The fixed integer package is particularly useful with 64 bit machines, and fairly large combinatorial problems (15-20 dimensions) have been correctly solved. It may be useful for doing empirical work, eg. searching for polyhedra with certain properties. Final results should always be checked using lrsmp. If you have GNU MP installed, "make gmp" will produce binaries glrs/gredund using this library. It may be necessary to edit the makefile to specify the path to gmp.
 

---

Estimation

The estimation feature of lrs allows estimates to be made of the output size and running time. These are based on Knuth's technique for estimating the size of backtracking trees, and are described in Avis and Devroye(1994).   The estimate is unbiased, that is the expected value of the estimate is the actual value. To get an estimate use the maxdepth option to limit the search depth, and the estimates option:

This will cause lrs to perform k random probes from each  node of the tree at depth d. k should be at least 1 and d at least zero.

H-representation: If the input is an H-representation, the program gives an unbiased estimate of the number of vertices and  rays in the V-representation,  and the total number of bases that will be computed by lrs. For the H-representation cube.ine, the options maxdepth 1 and estimates 1 produce the output:
*Estimates: vertices=9    rays=0    bases=9
*Total number of tree nodes evaluated: 6
In this case the V-representation of the cube is estimated to have 9 vertices, and it is estimated that lrs will compute a total of 9 bases. The estimate was based on evaluating 6 tree nodes.  Note: The estimate for the number of rays may be an overestimate if the polyhedron is not a cone, since some rays may be duplicated in the output - see subsection Output Duplication.

V-representation: If the input is a V-representation, the program gives an unbiased estimate of the number of facets in the H-representation, and the total number of bases that lrs will compute.  For V-representation cube.ext, the options maxdepth 0 and estimates 3 produce the output:
*Estimates:  facets=7  bases=8
*Total number of tree nodes evaluated: 7
In this cases it is estimated that the H-representation of the cube will contain 7 facets, and it is estimated that  lrs will compute a total of 7 bases to find it. The estimate was formed by evaluating 7 tree nodes. Note: The number of facets estimated may be an overestimate if the polyhedron is not bounded - see subsection Output Duplication.

Voronoi diagrams: Estimates for the number of Voronoi vertices and Voronoi rays for a V-representation of a set of data points may be obtained by combining the voronoi, estimates and maxdepth options.

Repeated estimates: In order to get estimates with different random probes, lrs can be given a seed for the random number generator. There are two ways: an option and a command line argument.

The integer n is used as a seed for the random number generator.

The command line argument is an integer n which will be the seed and overrides a seed given as an option.

Using the estimator: The running time of lrs is proportional to the number of bases, so an estimate of the number of bases gives an easy way to estimate the running time for solving the complete problem by lrs. For the case of polytopes that contain the origin, a V-representation can be processed as an H-representation and vice-versa (this is an application of duality). Hence facet estimates for a V-representation can also be obtained by running the problem as an H-representation with the estimates option. The estimated number of vertices will be in this case be an unbiased estimate for the number of facets for the original problem. So running the estimator on the file cube.ext with the header set as H-representation , maxdepth 0 and estimates 3, we get the output:
 *Estimates: vertices=5    rays=0    bases=9
*Total number of tree nodes evaluated: 8
Since the origin is an interior point, the estimated number of vertices is an accurate estimate of the number of facets of the H-representation of the cube.   Similarly, estimates for an input  H-representation of a polytope containing the origin may be obtained by processing the file as a V-representation. The output will be essentially the same, but the number of bases computed may be very different, see the subsection H- vs V-representation.   For a large problem of this type, it is useful to get estimates for the number of bases that lrs will compute for both V- and H-representations, so that simpler problem can be chosen.

The estimates may also be used to judge the feasibility of solving the problem using other codes. For example, any code that uses triangulation/perturbation to resolve degeneracy will have trouble if the number of bases is huge. Codes which must store all the output in memory (currently all codes except reverse search methods such as lrs) will have trouble if the estimated output size is huge.

Fourier Elimination

Nash Equilibria                   New in Version 4.2

  Linear Programming             ( Revised V4.2)

lrs can be used to solve linear programming problems in rational arithmetic when the input is an H-representation.  Use the option:

             and one of the options maximize or minize:

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

 simply maximizes the function  a₀ + a₁ x ₁ + ... + a_n-1 x_n-1
 over the given polyhedron. A optimal vertex is given when it exists, otherwise for unbounded solutions a vertex and ray is given. A minimization will be performed if the  following option is specified:

To print the dictionary at a few key points also include the option:

verbose

New in V4.2. Dual variables are now printed at termination. If the linearity option is used, only a partial list of dual variables will be given.
                       Dual variable y_i refers to inequality number i in the input.

Volume Computation

volume                                                                             // V-representation only //

will cause the volume to be computed. For input cube.ext, the output is:
*Volume=8
If the volume option is applied to an H-representation, the results are not predictable. If the option is applied to a V-representation of  a polytope that is not full dimensional, the volume of a projected polytope is computed. The projection used is to the lexicographically smallest coordinate subspace, see Avis, Fukuda, Picozzi (2002). 

For polytopes given by a H-representation, it will first be necessary to compute the V-representation.

Voronoi Diagrams

p₁  , p₂  , ...., p _n-1     ->    (p₁ ²  +   p₂² +  ...   +  p_n-1²  ) - 2 p₁  x ₁  - 2 p₂  x₂ - .... - 2  p _n-1 x_{n -1}  + x _n>= 0

lrs is applied to the H-representation so created.  This transformation is performed automatically for a V-representation if the

voronoi          // V-representation only - place immediately after end statement //

option is specified.
Note: The input file must consist entirely of data points (no rays), i.e.. there must be a one in column one of each line. The volume option should not be used, since the volume reported will not be the volume of the original V-representation.
The output will consist of the Voronoi vertices (columns beginning with a one) and Voronoi rays (columns beginning with zero) for the Voronoi diagram defined on the data points.  If the printcobasis option is given, the n "data points" indices produced will tell which set of input data points corresponds to the given Voronoi vertex or ray. In case of degeneracies, a given Voronoi vertex may be generated by more than n of the input data points. In this case, use of the allbases option will cause all  sets of n input data points corresponding to a Voronoi vertex to be printed. For Voronoi rays, the immediately preceding  cobasis is the cobasis of the the Voronoi vertex from which the ray emanates.  The index followed by a * is the data point to drop in order to generate the ray. If the geometric option is given the correspondence between Voronoi rays and Voronoi vertices will be produced automatically.

Example: Compute the Voronoi diagram of the planar point set (0,0), (2,1), (1,2), (0,4), (4,0), (4,4) (2,-4).
vor7-3
*6 Voronoi vertices and 5 rays
*7 input data points
V-representation
begin
7 3 integer
1 0 0
1 2 1
1 1 2
1 0 4
1 4 0
1 4 4
1 2 -4
end
voronoi
printcobasis
allbases
geometric

The output produced is
 

V-representation
begin
***** 3 rational
V#1 R#0 B#1 h=0 data points  1 5 7 det=64
1 2 -3/2
V#1 R#1 B#1 h=0 data points  1 5* 7 det=64
0 -2 -1  * 1 2 -3/2
V#1 R#2 B#1 h=0 data points  1* 5 7 det=64
0 2 -1  * 1 2 -3/2
V#1 R#2 B#2 h=1 data points  1 2 5 det=16
1 2 -3/2
V#2 R#2 B#3 h=2 data points  1 2 3 det=12
1 5/6 5/6
V#3 R#2 B#4 h=3 data points  1 3 4 det=16
1 -3/2 2
V#3 R#3 B#4 h=3 data points  1 3* 4 det=16
0 -1 0  * 1 -3/2 2
V#4 R#3 B#5 h=2 data points  2 5 6 det=32
1 15/4 2
V#4 R#4 B#5 h=2 data points  2* 5 6 det=32
0 1 0  * 1 15/4 2
V#5 R#4 B#6 h=3 data points  2 3 6 det=20
1 27/10 27/10
V#6 R#4 B#7 h=4 data points  3 4 6 det=32
1 2 15/4
V#6 R#5 B#7 h=4 data points  3* 4 6 det=32
0 0 1  * 1 2 15/4
end

The output contains 6 Voronoi vertices :
(2, -3/2), (5/6,5/6),(-3/2,2),(15/4,2), (27/10,27/10), (2,15/4).
The Voronoi vertex (2,-3/2) appears twice in the output with data point indices 1 5 7 and 1 2 5. This means that it is degenerate and is defined by the set of 4 input data point in positions 1,2,5,7 in the input file. I.e.. it is the centre of an empty  circle through the four input data points (0,0), (2,1), 4,0), (2,-4).  The other Voronoi vertices appear once each and are defined respectively by the data points with indices (i.e..  position in the input file)  1 2 3,  1 3 4,  2 5 6,  2 3 6 and 3 4 6. The  Voronoi diagram has 5 rays
(2, -3/2) + (-2t,-t),    (2,-3/2)+(2t,-t),    (-3/2,2)+(-t,0),    (15/4,2)+(t,0),    (2,15/4)+(0,t)

For example, the first ray in the output appears:
V#1 R#1 B#1 h=0 data points  1 5* 7 det=64
0 -2 -1  * 1 2 -3/2
 This means that the ray (-2t,-t) emanates from the vertex defined by data points 1 5 7, namely (2, -3/2). The asterisk on index 5 indicates that the ray is defined by the data points with indices 5 and 7, namely (0,0) and (2,-4).
 
 

---

  Extreme Point Enumeration and Redundant Inequalities

To remove input points that are not vertices from a V-representation or redundant inequalities from an H-representation use the command:

redund filename

Note: Versions of redund prior to this release failed to remove redundancy from the starting basis.

Linearities       (New in Version 4.0)

linearity  k  i₁ i₂ i ... i_k

The input file contains k linearities. If the input is a H-representation, the rows i₁ i₂ i ... i_k of the input file are equations. For a V-representation, the rows with these indices should begin with zero in column one, and will be interpreted as lines rather than rays.  Linearities defined on the input vertices of a V-representation are not defined, but the program will accept them and produce some output. Each of the indice i_k must be a distinct number between 1 and m. With an  H-representation, linearities are useful for enumeration of vertices on a facet or lower dimensional subspace. For example the file:

cube_ridge
*cube of side 2 centred at the origin
H-representation
linearity 2  1 5
begin
6 4 rational
1 1 0 0
1 0 1 0
1 0 0 1
1 -1 0 0
1 0 -1 0
1 0 0 -1
end

causes vertices to be enumerated on the ridge which is the intersection of the two facets

x₁ = -1   and   x₂ = 1

so the output is the pair of vertices

cube_ridge
*Input linearity in row(s) 1 5
V-representation
begin
2  4  rational
 1 -1  1  1
 1 -1  1 -1
end

Specifying linearities in this way will often produce redundancy , especially if the dimension of the problem is reduced considerably. As a preprocessing step, it is useful to apply to remove any redundancy by redund. In the case of the above problem the output produced by redund is:

cube
*Input linearity in row(s) 1 5
*row 2 was redundant and removed
*row 4 was redundant and removed
H-representation
linearity 2 1 2
begin
4 4 rational
 1  1  0  0
 1  0 -1  0
 1  0  0  1
 1  0  0 -1

and two redundant halfspaces were removed.

Redundant columns are closely related to linearities. If we examine the V-representation of cube_ridge above we can see that it is just a line segment in 3 dimensional space. Further,  columns 2 and 3 are multiples of column 1. If lrs is applied to this file, the column redundancies give rise to two linearities, so the output will appear as the H-representation given above: geometrically the intersection of two planes (the linearities) with two half-planes (defining the endpoints of the line segment).

In general, the representation of the linearity space is not unique, however the one produced by lrs should be the same as that produced by cdd.

Timing and Interrupts

Error Messages and Troubleshooting

The following error messages are produced by lrs . They are  arranged in alphabetic order.

Data type must be integer of rational

lrs does not handle floating point data, change to integer or rational input.

The digits option was specified, but the number of  decimal digits is too large (the values shown here is for 64 bit machines). MAX_DIGITS  (default 255) is the maximum number of array locations used for extended precision arithmetic. This value should be increased  and lrs recompiled.

The input polyhedron does not have dimension n-1. Either there is a mistake in the input, or it must be projected onto a full dimensional subspace. cdd has a function for doing this. The second message only appears for V-representations - note that at least one vertex must be supplied, or else the problem can be run as a H-representation.

An attempt to restart from an invalid cobasis has been made. Check that the indices are entered exactly as they appear in the previous aborted run.

LP operations can only be performed on H-representations.

Input file does not contain a begin line. See File Formats.

The input file is incorrect, please check File Formats.

The input is an H-representation of an infeasible system.

The startingcobasis option has been used, but the indices supplied were not valid: i.e. distinct numbers between 1..m.

An attempt to restart from an invalid cobasis has been made. Check that the indices are entered exactly  in the order appear in the previous aborted run.

---

Hints and Comments

H- vs V- representation

Redundancy vs Degeneracy

Even with redundant input removed a polyhedron may be highly degenerate. In directory ine/metric there are many highly degenerate combinatorial polytopes. These are difficult problems for all vertex enumeration/convex hull programs that use pivoting, such as lrs.  For example, the file ccc7.ine is a cone with 63 facets in 21 dimensions. It has 38,780 extreme rays, but computing these required the evaluation of 247,271,659 bases!

Memory considerations

Geometric Rays

For degenerate inputs, pivot based methods such as lrs may generate the same output vertex/ray/facet many times. Unless the allbases option is specified, lrs makes checks in order to remove duplicates.  An output is only printed when it occurs with a lexicographically minimum basis. This removes all duplicate vertices, but rays/facets may still be output more than once. This is due to the fact that duplicate geometric rays cannot always be detected. A warning message is produced when duplicates may occur in the output. They can be removed using the program buffer.c. Two important types of input never produce duplicate output: polytopes (i.e. bounded polyhedra) and cones (i.e. polyhedra where the origin is the only vertex).

---

Acknowledgements and References

D. Avis, lrs: A Revised Implementation of the Reverse Search Vertex Enumeration Algorithm, http://cgm.cs.mcgill.ca/~avis/doc/avis/Av98a.ps
   In: Polytopes - Combinatorics and Computation, Ed. G. Kalai and G. Ziegler, Birkhauser-Verlag (2000) 177-198.

D. Avis, "Computational Experience with the Reverse Search Vertex Enumeration Algorithm," Optimization Methods and Software, (1998 (to appear)). http://cgm.cs.mcgill.ca/~avis/doc/avis/Av98b.ps

D. Avis, D. Bremner, and R. Seidel, "How Good are Convex Hull Algorithms?," Computational Geometry: Theory and Applications, Vol 7,pp.265-301(1997). http://cgm.cs.mcgill.ca/~avis/doc/avis/ABS96a.ps

D. Avis and L. Devroye, "Estimating the Number of Vertices of a Polyhedron," pp. 179-190 in Snapshots of Computational and Discrete Geometry, ed. D. Avis and P. Bose, School of Computer Science, McGill University (1994). http://cgm.cs.mcgill.ca/~avis/doc/avis/AD94a.ps
  In: Information Processing Letters, (2000) V. 73, pp. 137-143.

D. Avis and K. Fukuda, "A Pivoting Algorithm for Convex Hulls and Vertex Enumeration of Arrangements and Polyhedra," Discrete and Computational Geometry, Vol. 8, pp. 295-313 (1992).  http://cgm.cs.mcgill.ca/~avis/doc/avis/AF92b.ps

D. Bremner, K. Fukuda and A. Marzetta, Primal-Dual Methods for Vertex and Facet Enumeration, 13th ACM  Symposium on Computational Geometry SCG 1997, 49-56.   http://www.cs.unb.ca/profs/bremner/pd/