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C++ Interface

Compile C++ sources and run tests

To compile the C++ code you need the lp_solve library. For example, for Unix/Linux you need liblpsolve55.so. This is available from the library's webpage as well as a package in several linux distributions e.g. debian sudo apt-get install lp-solve.

You have to specify the path to liblpsolve55.so/dll/dylib, by running, in folder test:

mkdir -p test/build && cd test/build
cmake -DLP_SOLVE=_PATH_TO_LIB_FILE_ ..
make

For example: -DLP_SOLVE=/usr/lib/lpsolve/liblpsolve55.so

You can run the tests by cmake test or ctest -jK where K the number of CPU threads. By adding the option --verbose to ctest you get more information about the tests, e.g. time per test, volume computed and the name of the polytope or convex body.

Development environment from Docker container

Optionally, it is possible to setup a docker contaner with development environment. To get started with docker, see here. Below, here is how a Dockerfile can be written

FROM ubuntu:18.04

RUN apt-get update && apt-get install -y g++ cmake lp-solve && \
    rm -rf /var/lib/apt/lists/*

Please, create Dockerfile.dev with above content inside a temporary folder (e.g. docker) and to create an image and run a container:

# build docker image
cd docker
docker build -t volesti:dev -f Dockerfile.dev .
# check built image
docker images | grep volesti
# run a container in an interactive mode from volesti source folder
docker run -it -v $PWD:/volesti -w /volesti --name=volesti-dev volesti:dev /bin/bash

Polytope input

The current version of the software assumes that the polytope is given in the form of linear inequalities i.e. {x \in R^d : Ax <= b} where A is a matrix of dimension m x d and b a vector of dimension m or as a set of m vertices {\in R^d} or as a Minkowski sum of m segments {\in R^d}. The input is described in an .ine-file (H-polytopes) or in a .ext file (V-polytopes or zonotopes). The .ine file is described as follows:

various comments  
begin  
m d+1 numbertype  
b -A  
end  
various options

The .ext file is described as follows:

various comments  
begin  
m d+1 numbertype  
1 v_1  
.. ...  
1 v_m  
end  
various options

In V-polytope case v_i are vertices and in zonotope case they are segments.

This filestype (or similar) is used by a number of other software in polyhedral computation (e.g. cdd, vinci, latte). In the current version of the software, the options are treated as comments and the numbertype as C++ double type.
If your input has equality constraints then you have to transform it in the form that only contain linear inequalities which described above by using some other software. We recommend to use latte https://www.math.ucdavis.edu/~latte for this transformation.

Run volesti from command line

After successful compilation you can use the software by command line. For example, the following command ./vol -h will display a help message about the program's available options.

Example

To estimate the volume of the 10-dimensional hypercube first prepare the file cube10.ine as follows:

cube10.ine  
H-representation  
begin  
 20 11 real  
 1 1 0 0 0 0 0 0 0 0 0  
 1 0 1 0 0 0 0 0 0 0 0  
 1 0 0 1 0 0 0 0 0 0 0  
 1 0 0 0 1 0 0 0 0 0 0  
 1 0 0 0 0 1 0 0 0 0 0  
 1 0 0 0 0 0 1 0 0 0 0  
 1 0 0 0 0 0 0 1 0 0 0  
 1 0 0 0 0 0 0 0 1 0 0  
 1 0 0 0 0 0 0 0 0 1 0  
 1 0 0 0 0 0 0 0 0 0 1  
 1 -1 0 0 0 0 0 0 0 0 0  
 1 0 -1 0 0 0 0 0 0 0 0  
 1 0 0 -1 0 0 0 0 0 0 0  
 1 0 0 0 -1 0 0 0 0 0 0  
 1 0 0 0 0 -1 0 0 0 0 0  
 1 0 0 0 0 0 -1 0 0 0 0  
 1 0 0 0 0 0 0 -1 0 0 0  
 1 0 0 0 0 0 0 0 -1 0 0  
 1 0 0 0 0 0 0 0 0 -1 0  
 1 0 0 0 0 0 0 0 0 0 -1  
end  
input_incidence

Then to use SequenceOfBalls (SOB) algorithm run the following command:

./vol -f1 cube_10.ine

which returns 17 numbers:
d m #experiments exactvolOr-1 approxVolume [.,.] #randPoints walkLength meanVol [minVol,maxVol] stdDev errorVsExact maxminDivergence time timeChebyshevBall

To use CoolingGaussian (CG) algorithm run the following command:

./vol -f1 cube_10.ine -cg

which returns the same output as before.

To estimate the volume of a 10-dimensional V-cross polytope described in cross_10.ext as follows:

cross_10.ext  
V-representation  
begin  
 20 11 integer  
 1 1 0 0 0 0 0 0 0 0 0  
 1 0 1 0 0 0 0 0 0 0 0  
 1 0 0 1 0 0 0 0 0 0 0  
 1 0 0 0 1 0 0 0 0 0 0  
 1 0 0 0 0 1 0 0 0 0 0  
 1 0 0 0 0 0 1 0 0 0 0  
 1 0 0 0 0 0 0 1 0 0 0  
 1 0 0 0 0 0 0 0 1 0 0  
 1 0 0 0 0 0 0 0 0 1 0  
 1 0 0 0 0 0 0 0 0 0 1  
 1 -1 0 0 0 0 0 0 0 0 0  
 1 0 -1 0 0 0 0 0 0 0 0  
 1 0 0 -1 0 0 0 0 0 0 0  
 1 0 0 0 -1 0 0 0 0 0 0  
 1 0 0 0 0 -1 0 0 0 0 0  
 1 0 0 0 0 0 -1 0 0 0 0  
 1 0 0 0 0 0 0 -1 0 0 0  
 1 0 0 0 0 0 0 0 -1 0 0  
 1 0 0 0 0 0 0 0 0 -1 0  
 1 0 0 0 0 0 0 0 0 0 -1  
end  
hull  
incidence

Run:

./vol -f2 cross_10.ext

which returns the same output as before.

To estimate the volume of a 4-dimensional zonotope defined by the Minkowski sum of 8 segments described in zonotope_4_8.ext as follows:

zonotope_4_8.ext  
Zonotpe  
begin  
 8 5 real  
 1 0.981851 -0.188734 -0.189761 0.0812645  
 1 -0.0181493 0.811266 -0.189761 0.0812645  
 1 -0.0181493 -0.188734 0.810239 0.0812645  
 1 -0.0181493 -0.188734 -0.189761 1.08126  
 1 -0.177863 0.437661 -0.0861379 -0.674634  
 1 0.737116 -0.204646 -0.540973 -0.471883  
 1 -0.684154 0.262324 0.292341 -0.265955  
 1 -0.802502 -0.740403 0.0938152 0.0874131  
end  
hull  
incidence

Run:

./vol -f3 zonotope_4_8.ext

Flag -v enables the print mode.

Generate polytopes

You can use executable generator to generate polytopes (hypercubes, simplices, cross polytopes, skinny hypercubes (only in H-representation), product of two simplices (only in H-representation) and zonotoes. For example:

  1. To generate a 10-dimensional hypercube in H-representation run:
./generate -cube -h -d 10
  1. To generate a 20-dimensional simplex in V-representaion run:
./generate -simplex -v -d 20
  1. To generate a 5-dimensional zonotope defined by 10 segments run:
./generate -zonotope -d 5 -m 10

Command ./generate -help will display a help message about the program's available options.

Sampling

You can sample from a convex polytope uniformly or from the spherical gaussian distribution. For example:

  1. To sample uniformly from the 10-dimensional hypercube, run:
./vol -f1 cube_10.ine -rand -nsample 1000

Flag -nsample declares the number of points we wish to sample (default is 100).

  1. To sample from the gaussian distribution, run:
./vol -f1 cube_10.ine -rand -nsample 1300 -gaussian -variance 1.5

Flag -variance declares the variance (default is 1.0). The center of the spherical gaussian is the Chebychev center for H-polytopes, or the origin for zonotopes. For V-polytopes is the chebychev center of the simplex that is defined by a random choice of d+1 vertices.

  1. To sample from a zonotope described in zonotope.ext file run:
./vol -f3 zonotope.ext -rand -nsample 1500

For V-polytopes use flag -f2 before the .ext file. In all cases use flag -v to print the excecutional time.