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# lrslib - Man Page

### Name

lrslib: Convert between representations of convex polyhedra, remove redundant inequalities, find minimum representations, convex hull computation, solve linear programs in exact precision,  compute Nash-equibria in 2-person games.

### Synopsis

lrs [input-file] [output-file]

redund [input-file] [output-file]

minrep [input-file] [output-file]

fel [input-file] [output-file]

mpirun -np num-proc mplrs input-file [output-file] [options]

lrsnash [options] [input-file]

hvref/xvref [input-file]

polyv [input-file]

### 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). lrslib is a C library containing programs to manipulate these representations. All computations are done in exact arithmetic.

lrs(1) converts an H-representation of a polyhedron to its V-representation and vice versa, known respectively as the vertex enumeration and facet enumeration problems (see Example (1) below). lrs can also be used to solve a linear program, remove linearities from a system, and extract a subset of columns.

redund(1) removes redundant inequalities in an input H-representation and outputs the remaining inequalities. For a V-representation input it outputs all extreme points and extreme rays. Both outputs can be piped directly into lrs. redund is a link to lrs which performs these functions via the redund and redund_list options.

minrep(1) performs the same functions as redund(1) but in addition searches for hidden linearities in the input. These are made explicit in the output which is a minimum representation of the polyhedron.

fel(1) projects an input H-representation onto a given set of variables using Fourier-Motzkin elimination. For a V-representation it extracts the specified columns. The output is non-redundant and can be  can be piped directly into lrs. fel is a link to lrs which performs these functions via  the eliminate and project options.

mplrs(1) is Skip Jordan's parallel wrapper for lrs/redund/minrep/fel.

lrsnash(1) is Terje Lensberg's application of lrs for finding Nash-equilibria in 2-person games.

hvref(1) xvref(1) produce a cross reference list between H- and V-representations.

polyv(1) is Skip Jordan's utility to create logical formulas for checking equivalence between H- and V- representations or determining whether a given inequality is redundant after eliminating variables, without eliminating the variables.

### Arithmetic

lrsarith(5) From version 7.1 lrs/redund/mplrs use hybrid arithmetic with overflow checking,  starting in 64bit integers, moving to 128bit (if available) and then GMP. Overflow checking is conservative to improve performance: eg. with 64 bit arithmetic, a*b triggers overflow if either a or b is at least 2^31,  and a+b triggers an overflow if either a or b is at least 2^62. Typically problems that can be solved in 64bits run 3-4 times faster than with GMP  and inputs solvable in 128bits run twice as fast as GMP.

Various arithmetic versions are available  and can be built from the makefile.

### Notes

User's guide for lrslib

### Author

David Avis <avis at cs dot mcgill dot ca >