Mathematical Programming

, Volume 110, Issue 1, pp 3–20

Optimizing over the first Chvátal closure

FULL LENGTH PAPER

DOI: 10.1007/s10107-006-0054-8

Cite this article as:
Fischetti, M. & Lodi, A. Math. Program. (2007) 110: 3. doi:10.1007/s10107-006-0054-8

Abstract

How difficult is, in practice, to optimize exactly over the first Chvátal closure of a generic ILP? Which fraction of the integrality gap can be closed this way, e.g., for some hard problems in the MIPLIB library? Can the first-closure optimization be useful as a research (off-line) tool to guess the structure of some relevant classes of inequalities, when a specific combinatorial problem is addressed? In this paper we give answers to the above questions, based on an extensive computational analysis. Our approach is to model the rank-1 Chvátal-Gomory separation problem, which is known to be NP-hard, through a MIP model, which is then solved through a general-purpose MIP solver. As far as we know, this approach was never implemented and evaluated computationally by previous authors, though it gives a very useful separation tool for general ILP problems. We report the optimal value over the first Chvátal closure for a set of ILP problems from MIPLIB 3.0 and 2003. We also report, for the first time, the optimal solution of a very hard instance from MIPLIB 2003, namely nsrand-ipx, obtained by using our cut separation procedure to preprocess the original ILP model. Finally, we describe a new class of ATSP facets found with the help of our separation procedure.

Keywords

Integer programs Separation problems Chvátal–Gomory cuts Computational analysis 

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.DEIUniversity of PadovaPadovaItaly
  2. 2.DEISUniversity of BolognaBolognaItaly

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