TY - GEN A1 - Beckenbach, Isabel A1 - Hatzel, Meike A1 - Wiederrecht, Sebastian T1 - The Tight Cut Decomposition of Matching Covered Uniformable Hypergraphs N2 - The perfect matching polytope, i.e. the convex hull of (incidence vectors of) perfect matchings of a graph is used in many combinatorial algorithms. Kotzig, Lovász and Plummer developed a decomposition theory for graphs with perfect matchings and their corresponding polytopes known as the tight cut decomposition which breaks down every graph into a number of indecomposable graphs, so called bricks. For many properties that are of interest on graphs with perfect matchings, including the description of the perfect matching polytope, it suffices to consider these bricks. A key result by Lovász on the tight cut decomposition is that the list of bricks obtained is the same independent of the choice of tight cuts made during the tight cut decomposition procedure. This implies that finding a tight cut decomposition is polynomial time equivalent to finding a single tight cut. We generalise the notions of a tight cut, a tight cut contraction and a tight cut decomposition to hypergraphs. By providing an example, we show that the outcome of the tight cut decomposition on general hypergraphs is no longer unique. However, we are able to prove that the uniqueness of the tight cut decomposition is preserved on a slight generalisation of uniform hypergraphs. Moreover, we show how the tight cut decomposition leads to a decomposition of the perfect matching polytope of uniformable hypergraphs and that the recognition problem for tight cuts in uniformable hypergraphs is polynomial time solvable. T3 - ZIB-Report - 18-61 KW - uniform hypergraphs KW - perfect matchings KW - tight cuts Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-71149 SN - 1438-0064 ER - TY - GEN A1 - Beckenbach, Isabel A1 - Borndörfer, Ralf A1 - Knoben, Loes A1 - Kretz, David A1 - Uetz, Marc T1 - The S-Bahn Challenge in Berlin N2 - The S-Bahn Challenge in Berlin is a challenge in which one has to travel the entire S-Bahn network of Berlin in the shortest possible time. We explain how the problem can be modeled and solved mathematically. Furthermore, we report on our record attempt on January 10, 2015. T3 - ZIB-Report - 15-13 Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-53792 UR - https://gor.uni-paderborn.de/index.php?id=362 SN - 1438-0064 ER - TY - GEN A1 - Beckenbach, Isabel A1 - Scheidweiler, Robert T1 - Perfect f-Matchings and f-Factors in Hypergraphs - A Combinatorial Approach N2 - We prove characterizations of the existence of perfect f-matchings in uniform mengerian and perfect hypergraphs. Moreover, we investigate the f-factor problem in balanced hypergraphs. For uniform balanced hypergraphs we prove two existence theorems with purely combinatorial arguments, whereas for non-uniform balanced hypergraphs we show that the f-factor problem is NP-hard. T3 - ZIB-Report - 16-22 KW - perfect f-matchings in hypergraphs KW - f-factors in hypergraphs KW - mengerian hypergraph KW - balanced hypergraph KW - perfect hypergraph KW - Hall's Theorem Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-59071 SN - 1438-0064 ER - TY - GEN A1 - Beckenbach, Isabel A1 - Peis, Britta A1 - Schaudt, Oliver A1 - Scheidweiler, Robert T1 - A Combinatorial Proof of a König-type Theorem for Unimodular Hypergraphs N2 - We state purely combinatorial proofs for König- and Hall-type theorems for a wide class of combinatorial optimization problems. Our methods rely on relaxations of the matching and vertex cover problem and, moreover, on the strong coloring properties admitted by bipartite graphs and their generalizations. T3 - ZIB-Report - 17-27 KW - Packing KW - Covering KW - Total Unimodularity KW - König's Theorem KW - Factors Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-64017 SN - 1438-0064 ER - TY - GEN A1 - Witzig, Jakob A1 - Beckenbach, Isabel A1 - Eifler, Leon A1 - Fackeldey, Konstantin A1 - Gleixner, Ambros A1 - Grever, Andreas A1 - Weber, Marcus T1 - Mixed-Integer Programming for Cycle Detection in Non-reversible Markov Processes N2 - In this paper, we present a new, optimization-based method to exhibit cyclic behavior in non-reversible stochastic processes. While our method is general, it is strongly motivated by discrete simulations of ordinary differential equations representing non-reversible biological processes, in particular molecular simulations. Here, the discrete time steps of the simulation are often very small compared to the time scale of interest, i.e., of the whole process. In this setting, the detection of a global cyclic behavior of the process becomes difficult because transitions between individual states may appear almost reversible on the small time scale of the simulation. We address this difficulty using a mixed-integer programming model that allows us to compute a cycle of clusters with maximum net flow, i.e., large forward and small backward probability. For a synthetic genetic regulatory network consisting of a ring-oscillator with three genes, we show that this approach can detect the most productive overall cycle, outperforming classical spectral analysis methods. Our method applies to general non-equilibrium steady state systems such as catalytic reactions, for which the objective value computes the effectiveness of the catalyst. T3 - ZIB-Report - 16-39 KW - Non-reversible Markov Processes KW - NESS KW - Mixed-Integer Programming KW - Markov State Models Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-60353 SN - 1438-0064 ER - TY - GEN A1 - Beckenbach, Isabel T1 - A Hypergraph Network Simplex Algorithm N2 - We describe a network simplex algorithm for the minimum cost flow problem on graph-based hypergraphs which are directed hypergraphs of a particular form occurring in railway rotation planning. The algorithm is based on work of Cambini, Gallo, and Scutellà who developed a hypergraphic generalization of the network simplex algorithm. Their main theoretical result is the characterization of basis matrices. We give a similar characterization for graph-based hypergraphs and show that some operations of the simplex algorithm can be done combinatorially by exploiting the underlying digraph structure. T3 - ZIB-Report - 17-36 Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-64539 SN - 1438-0064 ER - TY - GEN A1 - Beckenbach, Isabel A1 - Borndörfer, Ralf T1 - An Approximation Result for Matchings in Partitioned Hypergraphs N2 - We investigate the matching and perfect matching polytopes of hypergraphs having a special structure, which we call partitioned hypergraphs. We show that the integrality gap of the standard LP-relaxation is at most $2\sqrt{d}$ for partitioned hypergraphs with parts of size $\leq d$. Furthermore, we show that this bound cannot be improved to $\mathcal{O}(d^{0.5-\epsilon})$. T3 - ZIB-Report - 14-30 Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-51083 ER - TY - GEN A1 - Beckenbach, Isabel A1 - Borndörfer, Ralf T1 - A Hall Condition for Normal Hypergraphs N2 - Conforti, Cornuéjols, Kapoor, and Vusković gave a Hall-type condition for the non-existence of a perfect matching in a balanced hypergraph. We generalize this result to the class of normal hypergraphs. T3 - ZIB-Report - 15-45 KW - Matchings in hypergraphs KW - Normal hypergraphs KW - Hall condition Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-56175 SN - 1438-0064 ER -