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Integer programming proofs for Chvátal's conjecture over finite ground sets

Chvátal's Conjecture states that there can exist no intersecting family in a downset that has more elements than a maximal-sized star. A collection of work on the subject is available on Chvátal's Website. We present a computational framework that is able to prove the conjecture for ground sets of seven or fewer elements.

Leon Eifler, Ambros Gleixner, and Jonad Pulaj: A Safe Computational Framework for Integer Programming applied to Chvátal's Conjecture. Preprint.

This repository provides:

  • MPS and ZIMPL files that model the Integer Programs described in the paper.
  • Certificate files that show the correctness of the computational proofs
  • Links to external software
  • An input-checker written in Coq.

Solving Mixed Integer Programs Exactly

Exact SCIP is the exact rational variant of SCIP. A version that supports automatic certificate generation is available on the SCIP website. It interfaces QSopt_Ex as an exact rational LP solver. There is a small patch-file in this repository to enable correct printing of certificates in QSopt_Ex. The correctness of exact SCIP can be verified using VIPR, an independant certificate format for integer programming results.

An Input checker for VIPR certificates for Chvàtal's Conjecture

This small program written with the Coq proof assistant checks the input for certain VIPR certificate files that certify correctness for Chàtal's Conjecture for small ground sets, as proposed in

reference

This repository contains source code, several compilation scripts, as well as some example certificate files.

Installation instructions

This program needs a working installation of coq. It uses the Coq.io library which can be installed with opam via

  opam repo add coq-released https://coq.inria.fr/opam/released
  opam install -j4 -v coq-io-system

To create the Makefile simply run:

    ./configure.sh

Then run

 make && ./build.sh

to create the executable binary that does the typechecking. If you run the binary it will ask you for the filename of a vipr-file that should be checked, as well as the size of the groundset as input.

Use the python-script transvipr.py on a vipr file to rename the variables so that the order is correct. This does not change the validity of the VIPR certificate.