Showing papers by "Marc E. Pfetsch published in 2006"

Computing Optimal Morse Matchings

Abstract: Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is \NP-hard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computational results.

Orbitopal Fixing

Abstract: The topic of this paper are integer programming models in which a subset of 0/1-variables encode a partitioning of a set of objects into disjoint subsets Such models can be surprisingly hard to solve by branch-and-cut algorithms if the order of the subsets of the partition is irrelevant, since this kind of symmetry unnecessarily blows up the search tree We present a general tool, called orbitopal fixing, for enhancing the capabilities of branch-and-cut algorithms in solving such symmetric integer programming models We devise a linear time algorithm that, applied at each node of the search tree, removes redundant parts of the tree produced by the above mentioned symmetry The method relies on certain polyhedra, called orbitopes, which have been introduced bei Kaibel and Pfetsch (Math Programm A, 114 (2008), 1-36) It does, however, not explicitly add inequalities to the model Instead, it uses certain fixing rules for variables We demonstrate the computational power of orbitopal fixing at the example of a graph partitioning problem

Routing in Line Planning for Public Transport

Abstract: The line planning problem is one of the fundamental problems in strategic planning of public and rail transport. It consists in finding lines and corresponding frequencies in a network such that a given demand can be satisfied. There are two objectives. Passengers want to minimize travel times, the transport company wishes to minimize operating costs. We investigate three variants of a multi-commodity flow model for line planning that differ with respect to passenger routings. The first model allows arbitrary routings, the second only unsplittable routings, and the third only shortest path routings with respect to the network. We compare these models theoretically and computationally on data for the city of Potsdam.

Optimal Fares for Public Transport

Abstract: The fare planning problem for public transport is to design a system of fares that maximize the revenue. We introduce a nonlinear optimization model to approach this problem. It is based on a discrete choice logit model that expresses demand as a function of the fares. We illustrate our approach by computing and comparing two different fare systems for the intercity network of the Netherlands.

Competitive online multicommodity routing

Abstract: In this paper we study online multicommodity minimum cost routing problems in networks, where commodities have to be routed sequentially. The flow of each commodity can be split on several paths. Arcs are equipped with load dependent price functions defining routing costs. We discuss a greedy online algorithm that routes each commodity by minimizing a convex cost function that only depends on the demands previously routed. We present a competitive analysis of this algorithm showing that for affine linear price functions this algorithm is $\tfrac{4K}{2+K}$-competitive, where K is the number of commodities. For the parallel arc case, this algorithm is optimal. Without restrictions on the price functions and network, no algorithm is competitive. Finally, we investigate a variant in which the demands have to be routed unsplittably.

Packing and Partitioning Orbitopes

Abstract: We introduce orbitopes as the convex hulls of 0/1-matrices that are lexicographically maximal subject to a group acting on the columns. Special cases are packing and partitioning orbitopes, which arise from restrictions to matrices with at most or exactly one 1-entry in each row, respectively. The goal of investigating these polytopes is to gain insight into ways of breaking certain symmetries in integer programs by adding constraints, e.g., for a well-known formulation of the graph coloring problem. We provide a thorough polyhedral investigation of packing and partitioning orbitopes for the cases in which the group acting on the columns is the cyclic group or the symmetric group. Our main results are complete linear inequality descriptions of these polytopes by facet-defining inequalities. For the cyclic group case, the descriptions turn out to be totally unimodular, while for the symmetric group case both the description and the proof are more involved. Nevertheless, the associated separation problem can be solved in linear time also in this case.