@misc{DittelFuegenschuhMartin2011, author = {Dittel, Agnes and F{\"u}genschuh, Armin and Martin, Alexander}, title = {Polyhedral Aspects of Self-Avoiding Walks}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-12576}, number = {11-11}, year = {2011}, abstract = {In this paper, we study self-avoiding walks of a given length on a graph. We consider a formulation of this problem as a binary linear program. We analyze the polyhedral structure of the underlying polytope and describe valid inequalities. Proofs for their facial properties for certain special cases are given. In a variation of this problem one is interested in optimal configurations, where an energy function measures the benefit if certain path elements are placed on adjacent vertices of the graph. The most prominent application of this problem is the protein folding problem in biochemistry. On a set of selected instances, we demonstrate the computational merits of our approach.}, language = {en} } @misc{KaibelStephan2007, author = {Kaibel, Volker and Stephan, R{\"u}diger}, title = {On cardinality constrained cycle and path polytopes}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10093}, number = {07-25}, year = {2007}, abstract = {We consider polytopes associated with cardinality constrained path and cycle problems defined on a directed or undirected graph. We present integer characterizations of these polytopes by facet defining linear inequalities for which the separation problem can be solved in polynomial time. Moreover, we give further facet defining inequalities, in particular those that are specific to odd/even paths and cycles.}, language = {en} } @misc{Stephan2008, author = {Stephan, R{\"u}diger}, title = {Cardinality Constrained Combinatorial Optimization: Complexity and Polyhedra}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11026}, number = {08-48}, year = {2008}, abstract = {Given a combinatorial optimization problem and a subset \$N\$ of natural numbers, we obtain a cardinality constrained version of this problem by permitting only those feasible solutions whose cardinalities are elements of \$N\$. In this paper we briefly touch on questions that addresses common grounds and differences of the complexity of a combinatorial optimization problem and its cardinality constrained version. Afterwards we focus on polytopes associated with cardinality constrained combinatorial optimization problems. Given an integer programming formulation for a combinatorial optimization problem, by essentially adding Gr{\"o}tschel's cardinality forcing inequalities, we obtain an integer programming formulation for its cardinality restricted version. Since the cardinality forcing inequalities in their original form are mostly not facet defining for the associated polyhedra, we discuss possibilities to strengthen them.}, language = {en} }