Mathematical Programming

, Volume 124, Issue 1–2, pp 45–68 | Cite as

Small Chvátal Rank

  • Tristram BogartEmail author
  • Annie Raymond
  • Rekha Thomas
Full Length Paper Series B


We propose a variant of the Chvátal-Gomory procedure that will produce a sufficient set of facet normals for the integer hulls of all polyhedra {x : A x ≤ b} as b varies. The number of steps needed is called the small Chvátal rank (SCR) of A. We characterize matrices for which SCR is zero via the notion of supernormality which generalizes unimodularity. SCR is studied in the context of the stable set problem in a graph, and we show that many of the well-known facet normals of the stable set polytope appear in at most two rounds of our procedure. Our results reveal a uniform hypercyclic structure behind the normals of many complicated facet inequalities in the literature for the stable set polytope. Lower bounds for SCR are derived both in general and for polytopes in the unit cube.


Integer programming Cutting planes Chvátal rank Supernormality Stable set polytopes 

Mathematics Subject Classification (2000)

90C10 90C27 90C57 


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© Springer and Mathematical Programming Society 2010

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsQueen’s UniversityKingstonCanada
  2. 2.Berlin Mathematical SchoolTechnical UniversityBerlinGermany
  3. 3.Department of MathematicsUniversity of WashingtonSeattleUSA

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