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Algorithms for finding tight UTVPI (octagonal) over-approximations of polyhedra using linear programming and Fourier-Motzkin elimination
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FM and LP based UTVPI Over Approximations

Implements algorithms to compute tight UTVPI overapproximations of polyhedra.

The algorithms provided are:

  • Based on linear programming - LP0
  • Based on Fourier-Motzkin elimination - FM1, FM2

Two implementations of LP0 have been provided, one using CPLEX and the other using PIPLib. This code also includes a C++ API (in include/utvpi_oa_fm.h) with functionality for performing Fourier-Motzkin elimination (similar to FMLib) and using it to compute UTVPI overapproximations, and this has been used to implement FM1 and FM2.

Details of the algorithms can be found in the paper:

A. A. Patwardhan, P. Kumar, S. Rao, A. Miné and R. Upadrasta. On Tightest Octagonal Over-Approximations of Polyhedra.



  • CMake v3.12+
  • Compiler for C and C++
  • IBM CPLEX (for CPLEX version of LP0)
  • PIPLib (for PIPLib version of LP0)

Install cplex at $HOME/ibm/ILOG/CPLEX_Studio128.


To install FM1, FM2, and the CPLEX version of LP0, use:

$ mkdir build && cd build
$ cmake ..
$ make

To install the PIPLib version of LP0, use:

$ cd lp-pip
$ mkdir build && cd build
$ cmake ..
$ make

The path of the PIPLib installation can be passed to CMake using the PIPLIB_PATH variable.


The input polyhedron must be provided in the PolyLib/FMLib format. Example input files have been provided under the examples/ directory. A python script,, has also been provided to convert polyhedra in the H-representation in the cddlib format to the FMLib format.

Running the program

The program accepts a polyhedron as input from the standard input and sends its UTVPI overapproximation as output to the standard output.

The executable (named utvpi-oa for FM1/FM2/CPLEX LP0 and lp-pip for PIPLib LP0) can be found after make in the build directory.


This code is provided under the BSD 3-Clause License.


The implementation of the Fourier-Motzkin elimination functionality is inspired by FMLib.

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