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

  • What's New in Version 4.2
  •  What's New in Version 4.1
  • What's New in Version 4.0
  • Introduction  (revised)
  • Installation    (revised)
  • File Formats
  • Basic Options  (revised)
  • Arithmetic Packages 
  • Estimation
  •  Fourier Elimination (new in V4.2)
  • Linear Programming(revised)
  •  Nash Equilibria (new in V4.2)
  • Volume Computation
  • Voronoi Diagrams
  • Extreme Point Enumeration and Redundant Inequalities
  • Linearities
  • Timing and Interrupts
  • Error Messages and Troubleshooting
  • Hints and Comments
  • Acknowledgements and References

    What's New in Version 4.2

  • The main additions in Version 4.2 are two new drivers:

    There is a new bound option in lrs that truncates the reverse search tree when the objective function is less than the given bound for a maximization problem, or greater than the bound for minimization. When lrs is used to solve LPs, the dual variables will now be printed for the optimum solution.

    Other new basic options: dualperturb, project, printslack
    The nonnegative option should now work correctly  even if the origin is not a vertex of the polyhedron.

    What's New in Version 4.1

    Version 4.1 of lrslib contains no substantial revisions to the basic algorithms, but contains complete garbage collection and some simple drivers for solving vertex enumeration, convex hull and linear programming problems on generate polyhedra. The generation of input internally is simplified by the inclusion of some lpsolve like procedures to enable easy construction of the constraint matrix. Documentation for this is contained in http://cgm.cs.mcgill.ca/~avis/C/lrslib/lrslib.html

    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

    Note: The origin must be a vertex of the feasible region for Version 4.1.

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

    Version 4.0 of lrs is distributed as the callable library lrslib. Two drivers are supplied, lrs.c and redund.c , that replace the corresponding programs in previous distributions. The new programs should be compatible with earlier ones, and correctly process existing input files. New in this version is that input files need not be of full dimension, and input equations or linearities may be specified. In addition, input files may contain redundant columns, which are closely connected to linearities. See also the section File Formats for more information.

    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.


    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   contains a more detailed introduction to the problem, along with many useful tips for the new user.

    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 methodcdd 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:

    A second program, redund, is supplied for removing redundancy from and H or V-representation. In the former case, this involves the removal of any inequalities that are not required to represent the polyhedron. In the latter case, this is  the problem of evaluating the extreme points and rays of a V-representation. These problems are normally  considerably easier than the transforamtions performed by lrs. In some cases, redundancy can greatly slow the processing time taken by lrs, and it is advisble to remove any redundancy from the input file using redund before applying lrs.

    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.


    File Formats

    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

    New for Version 4.2: bound, dualperturb, printslack, project
    allbases bound  x                             New in Ver 4.2           // 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 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// 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 geometric                 // H-representation  or voronoi option only //               This indicates ray  0   r0   r 1 ...   rn-1is incident with vertex  1   v0   v 1 ...   vn-1.  For the quadrant, the output is: incidence                                         // New in version 4.0 //
    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.
    linearity  k  i1 i2 i ... ik
    The input contains k linearities in rows i1 i2 i ... ik of the input file are equations. See Linearities.
    maxdepth k 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 + a1 x 1 + ... + an-1 xn-1.
    The dualperturb option may be needed to avoid dual degeneracy.
    See Nash Equilibria and  Linear Programming
     mindepth k nonnegative        //modified in lrs 4.2 //                 // 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 lrs4.0

    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.

    Used by driver fourier only.

    restart  V# R# B# depth {facet #s or vertex/ray #s}                 Modified in lrs4.0 startingcobasis i1 i2 i ... in-1 truncate                                            // H-representation only //         
    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.
    volume                                              // V-representation  only // voronoi                                              // V-representation  only - place immediately after end statement //

    Arithmetic Packages                      (new in Version 4.0)

    Version 4.0 comes with a choice of arithmetic packages. Currently available are:

       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.


    maxdepth d
    estimates k seed n lrs n < filename

    Fourier Elimination

    Tallman Nkgau has contributed a driver fourier that computes the projection of a polyhedron given by its H-representation onto a selected set of coordinates. The program can be compiled by

    make fourier

    and is run by the command

    fourier file.ine [fileout]

    is a standard H-representation that does not contain linearities.
    Following the end statement, insert the option:

    project t  a1 ... at

    The output will be the polyhedron projected onto the coordinates a1 ... a , each of which is a unique number betwen 1 and n.

    Nash Equilibria                   New in Version 4.2

    All Nash equilibria (NE) for a two person noncooperative game are computed using two interleaved reverse search vertex enumeration steps.
    The input for the problem are two m by n matrices A,B of integers or rationals. The first player is the row player, the second is the column player.
    If row i and column j are played, player 1 receives Ai,j and player 2 receives Bi,j. (Program runs faster if m is <= n , see below.)

    The easiest way to use the program nash is to first run setupnash on a file containing:
     m n
    matrix A
    matrix B

    eg. the file game is for a game with m=3 n=2:
    3 2

    0 6
    2 5
    3 3

    1 0
    0 2
    4 3

    % setupnash game gane1 game2

    produces two H-representations, game1 and game2, one for each player.

    To get the equilibrium, run

    % nash game1 game2

    *nash:lrslib v.4.2, 2005.6.1(32bit,lrsmp.h)
    *Copyright (C) 1995,2005, David Avis   avis@cs.mcgill.ca
    *Input taken from file game1
    *Second input taken from file game2

    ***** 5 4 rational
    2  1/3  2/3  4
    1  2/3  1/3  0  2/3

    2  2/3  1/3  3
    1  0  1/3  2/3  8/3

    2  1  0  3
    1  0  0  1  4

    *Number of equilibria found: 3

    Each row beginning 1 is a strategy for the row player yielding a NE with each row beginning 2 listed immediately above it.
    The payoff for player 2 is the last number on the line beginning 1, and vice versa.
    Eg: first two lines of output: player 1 uses row probabilities 2/3 2/3 0 resulting in a payoff of 2/3 to player 2.
    Player 2 uses column probabilities 1/3 2/3 yielding a payoff of 4 to player 1.

    Optionally, lower bounds on the payoff functions to either or both players may be included.
    To give a lower bound of r on the payoff for player 1 add the options to file game2  (yes that is correct!)
    To give a lower bound of r on the payoff for player 2 add the options to file game1

    maximize 0 0 .... 0 1                                    (n entries to be given)
    bound    r                                                    ( r is integer or rational)

    If m is greater than n then the program usually runs faster by transposing the players. This is achieved by running:

    % nash game2 game1

    If you wish to construct the game1 and game2 files by hand, they are fragile and should be done exactly as follows:

       For player 1: eg. game1
       One linearity in the last row
       Identity matrix with additional final column 0
       Transpose of payoff matrix for player 2 with final column 1
       Last row is prob sum to one

       For player 2: eg. game2
       One linearity in the last row
       Payoff matrix for player 1 with final column 1
       Identity matrix with additional final column 0
       Last row is prob sum to one

    Corresponding to file game above we get

    *game: player 1
    linearity 1 6
    6 5 rational
    0 1 0 0 0
    0 0 1 0 0
    0 0 0 1 0
    0 -1 0 -4  1
    0 0 -2 -3  1
    -1 1 1 1 0

    *game: player 2
    linearity 1 6
    6 4 rational
    0 0 -6  1
    0 -2 -5  1
    0 -3 -3  1
    0 1 0 0
    0 0 1 0
    -1 1 1 0

      Linear Programming             ( Revised V4.2)


                 and one of the options maximize or minize:

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

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

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


    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 yi refers to inequality number i in the input.

    Volume Computation

    lrs can be used to compute the volume of a full dimensional polytope given as a V-representation. The option

    volume                                                                             // V-representation only //

    will cause the volume to be computed. For input cube.ext, the output is:
    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

    lrs can be used the compute the V-vertices of a Voronoi diagram of a set of data points in n-1 dimensional space. To do this we use a standard lifting procedure (see, e.g., Edelsbrunner, "Algorithms in Combinatorial Geometry," pp 296-297) . Each point is mapped to a half space tangent to the parabaloid in n dimensions, by the mapping:

    p , p , ...., p n-1     ->    (p1 +   p22 +  ...   +  pn-1 ) - 2 p1  x - 2 p x2 - .... - 2  p n-1 xn -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).
    *6 Voronoi vertices and 5 rays
    *7 input data points
    7 3 integer
    1 0 0
    1 2 1
    1 1 2
    1 0 4
    1 4 0
    1 4 4
    1 2 -4

    The output produced is

    ***** 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

    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

    A convex hull problem that occurs frequently is to enumerate the extreme points (vertices) of a given set of input points. This problem is in fact much simpler than the problem of finding the facets of the given input point set. It can be solved by linear programming.  The dual problem is to remove redundant inequalities from an H-representation. An input  inequality is redundant if it can be deleted without changing the polyhedron. The program redund also solves these problems. They can be solved by cdd using the vertex_listing  and facet_listing  options.

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

    The resulting file can be used directly with lrs, or even piped into lrs. In fact, lrs works best if the input is non-redundant, see the section Redundancy vs Degeneracy.

    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 ... ik

    The input file contains k linearities. If the input is a H-representation, the rows i1 i2 i ... ik 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 ik 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 of side 2 centred at the origin
    linearity 2  1 5
    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

    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

    *Input linearity in row(s) 1 5
    2  4  rational
     1 -1  1  1
     1 -1  1 -1

    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:

    *Input linearity in row(s) 1 5
    *row 2 was redundant and removed
    *row 4 was redundant and removed
    linearity 2 1 2
    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

    lrs handles certain signals unless it is compiled with the -DOMIT_SIGNALS option. It is possible to interrupt lrs and get the latest cobasis, which can be used for restarting the program (useful if the machine is going down!)
            signal                     operation
            USR1                         print current cobasis and continue
            TERM                       print current cobasis and terminate
            INT (ctrl-C)                      ditto
            HUP                                      ditto
    lrs also provides timing information, unless compiled with the option -DOMIT_TIMES.

    Error Messages and Troubleshooting

    The most common error occurs from an incorrect input file specification, please check the section File Formats carefully. In particular, lrs does not check the type or number of input coefficients specified.  After the line
    m n rational
    you must specify exactly m*n rational or integer coefficients. They are read  in free format , but normally each input facet or vertex/ray is begun on a new line.  See note for cdd users.

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

    Data type must be integer of rational

    Digits must be at most 2295  Change MAX_DIGITS and recompile Input Polyhedron does not have full dimension
    If input is a cone, change to H-representation, or add the origin 1 0 0 ... 0 Invalid Co-basis - does not have correct rank Maximize/minimize only valid for H-representation No begin line No data in file No feasible solution Starting cobasis indices must be distinct and in range 1 .. m Trying to restart from infeasible dictionary

    Hints and Comments

    H- vs V- representation

     lrs is programmed to manipulate H-representations directly. A file presented as a V-representation is processed by lifting it to a cone in one higher dimension, which is treated internally as a H-representation. If the input file is a polytope which contains the origin, then the user has two options. Submit it as a V-representation and have it processed as just described, or submit it as a H-representation, and interpret the output as a list of facet inequalities rather than "vertices". Since this will not be lifted, it will be processed in a different way by lrs. Sometimes a degenerate V-representation may run more quickly as a H-representation, and sometimes more slowly. To decide which representation to use for a large problem, the user can run the estimates option and choose the representation with fewest estimated bases.

    Redundancy vs Degeneracy

    For an H-representation, an input is redundant if some inequality can be deleted without changing the polyhedron. It is degenerate if (in d dimensions) at least one vertex lies on d+1 or more facets.  Similarly in a V-representation an input is redundant if some input point is not a vertex of the convex hull.  It is degenerate if some facet contains d+1 or more input points. The options   printcobasis and incidence give degeneracy information. Degeneracy causes pivot  or triangulation based methods such as lrs to  run slowly. Redundancy is one cause of degeneracy, but it can be avoided by pre-processing the input files. See section Extreme Point Enumeration and Redundant Inequalities for instructions on how to do this. This pre-processing is unnecessary if it is known that the input is non-redundant.

    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

    The strong point of lrs is that it does not save the output produced, so in theory it cannot run out of memory.  With cache size one all memory is allocated at the beginning, so if lrs starts running it will not run out of memory. It is possible however that the number of digits required to do the calculations exceeds the amount specified on the digits option, or the default. In practice, this problem will also arise early in the computation. In any case, a message is printed and the calculation can be restarted. In order to improve performance, some dictionaries should be cached. The default of 10 can be overridden by the cacheoption. If the dictionary is in the cache it does not need to be recomputed when backtracking, reducing  processing time by about 40%. Since the cache is allocated dynamically, a cache size that is too large can potentially use up large ammounts of machine memory.

    Geometric Rays

    A minimum V-representation of a polyhedron is a minimum set of vertices and rays such that each point in the polyhedron can be expressed as a convex combination of vertices plus a non-negative combination of rays. For the cube, if we delete the inequality
    x3 <= 1, i.e.. the line 1 0 0 -1 from file cube.ine, we get the output:
    ***** 4 rational
    1 1 1 -1
    0 0 0 1
    1 -1 1 -1
    1 1 -1 -1
    1 -1 -1 -1
    indicating the polyhedron is the convex combination of 4 vertices and 1 ray. With the geometric option, we get the output:
    ***** 4 rational
    1 1 1 -1
    0 0 0 1  * 1 1 1 -1
    1 -1 1 -1
    0 0 0 1  * 1 -1 1 -1
    1 1 -1 -1
    0 0 0 1  * 1 1 -1 -1
    1 -1 -1 -1
    0 0 0 1  * 1 -1 -1 -1
    This indicates that geometrically, the polyhedron has 4 parallel extreme rays (0,0,t) , one incident to each vertex. With the geometric option, all rays will be printed. Without the option, lrs tries to print each ray once, but in some cases duplicates will remain, see  subsection Output Duplication.

    Output Duplication

    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

    I would like to thank many people for helping with this implementation project. Komei Fukuda encouraged me from the start, collaborated in designing the file formats, and provided many suggestions for improving the code. Debugging would have been almost impossible without the use of his program cdd as a benchmark. David Bremner implemented memory allocation, cacheing and signals. Ambros Marzetta demonstrated the importance of cacheing and lrslong is based on his earlier implementation of this as prs_single.  Jerry Quinn coded the integer divide routine. Bug reports were provided by many users, for which I thank them. In particular Gerardo Garbulsky's extensive use of earlier versions suggested many refinements and Andreas Enge helped debug the volume computation. Tallman Nkgau contributed fourier.

    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. Avis, K. Fukuda and S. Picozzi, "On Canonical Representations of Convex Polyhedra", Mathematical Software,  ICMS 2002, Ed. A. Cohen, X-S Gao, N. Takayama, World Scientific, pp.350-360 (2002)   http://cgm.cs.mcgill.ca/~avis/doc/avis/AFP02a.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/