@misc{Groetschel, author = {Gr{\"o}tschel, Martin}, title = {Cardinality Homogeneous Set Systems, Cycles in Matroids, and Associated Polytopes}, doi = {10.1137/1.9780898718805.ch8}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6868}, number = {02-19}, abstract = {A subset \${\cal C}\$ of the power set of a finite set \$E\$ is called cardinality homogeneous if, whenever \${\cal C}\$ contains some set \$F\$, \${\cal C}\$ contains all subsets of \$E\$ of cardinality \$|F|\$. Examples of such set systems \${\cal C}\$ are the sets of circuits and the sets of cycles of uniform matroids and the sets of all even or of all odd cardinality subsets of \$E\$. With each cardinality homogeneous set system \${\cal C}\$, we associate the polytope \$P({\cal C})\$, the convex hull of the incidence vectors of all sets in \${\cal C}\$, and provide a complete and nonredundant linear description of \$P({\cal C})\$. We show that a greedy algorithm optimizes any linear function over \$P({\cal C})\$, give an explicit optimum solution of the dual linear program, and provide a polynomial time separation algorithm for the class of polytopes of type \$P({\cal C})\$.}, language = {en} }