Cardinality Constrained Combinatorial Optimization
To be more specific, we denote a combinatorial optimization problem by the triple Π = (E, I, w), where E is a finite set, I is a subset of the power set of E, called feasible solutions, and w is the objective function. The goal is to find a feasible solution F maximizing (minimizing) w. Moreover, let c = (c_{1}, ..., c_{m}) be a finite sequence of integers 0 ≤ c_{1} < c_{2} < ... < c_{m} ≤  E  Then, the optimization problem Π_{c}: max w(F), F ∈ I,  F  = c_{p} for some p is a cardinality constrained version of Π. Denote by P^{c} the polytope associated with Π_{c}, that is, the convex hull of the incidence vectors of feasible solutions with respect to Π_{c}. The aim of this project is a better understanding of facet defining inequalities that are specific to these cardinality conditions.
IPformulations
If one has an integer programming formulation of Π = (E, I, w), say Ax ≤ b, x_{e} ∈ {0,1} for all e ∈ E, then one obtains an integer programming formulation for Π_{c} by adding the cardinality bounds x(E) ≥ c_{1} and x(E) ≤ c_{m} as well as the forbidden set inequalities
The Cardinality Constrained Matroid Polytope
A basic idea for strenghtening forbidden set inequalities is not to consider the cardinality of a set F itself but the maximum size of the intersection of a feasible solution and F. When (E,I) is a matroid, then this is exactly the rank of F. One can show that the rank induced forbidden set inequalities
Poster
 Polyhedral Investigation of Cardinality Constrained Combinatorial Optimization Problems (05/2008) [(pdf)]
Publications
2010 

Volker Kaibel, Rüdiger Stephan  On cardinality constrained cycle and path polytopes  Math. Program., 123(2 (A)), pp. 371394, 2010 (preprint available as ZIBReport 0725) 
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2009 

Rüdiger Stephan  Facets of the (s,t)ppath polytope  Discrete Appl. Math., 157(14), pp. 31193132, 2009 (preprint available as ZIBReport 0638) 
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BibTeX DOI 
Jean Maurras, Rüdiger Stephan  On the cardinality constrained matroid polytope  2009 (preprint available as ZIBReport 0808) 
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BibTeX 
2005 

Rüdiger Stephan  Polytopes associated with length restricted directed circuits  diploma thesis, Technische Universität Berlin, 2005 
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2004 

Martin Grötschel  Cardinality Homogeneous Set Systems, Cycles in Matroids, and Associated Polytopes  The Sharpest Cut, Martin Grötschel (Ed.), MPSSIAM, pp. 99120, 2004 (preprint available as ) 
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