Frank Wagner

A simple min-cut algorithm

We present an algorithm for finding the minimum cut of an undirected edge-weighted graph. It is simple in every respect. It has a short and compact description, is easy to implement, and has a surprisingly simple proof of correctness. Its runtime matches that of the fastest algorithm known. The runtime analysis is straightforward. In contrast to nearly all approaches so far, the algorithm uses no flow techniques. Roughly speaking, the algorithm consists of about |V| nearly identical phases each of which is a maximum adjacency search.

A packing problem with applications to lettering of maps

The following packing problem arises in connection with lettering of maps: Given n distinct points pl, p2, . . . . pn in the plane, determine the supremum uoPi of all reals U, such that there are n pan-wise dtsjomt, axis-parallel, closed squares Ql, Q2, . . . . Qn of side-length u, where each pi ts a corner of Qi. Note that — by using afine transformation — the problem is equivalent to the case when we want largest homothetic cop~es of a jized rectangle or parallelogram tnstead of equal ly-szzed squares. In the cartographic application, the points are items (groundwater-drillho les etc.) and the squares are places for labels associated with these items (sulphate concentration etc.). An algorithm is presented, that in O(n log n] time either produces a solution, that is guaranteed to be at least half as large as the supremum. This is optimal, m the sense that the corresponding decision problem is NP complete, no po[ynomzal approximation algorithm with a guaranteed factor ezceedmg ~ exwts, provided that P # AfP; and there M also a lower bound of C2(n log n) for the running time.

Between Min Cut and Graph Bisection

We investigate a class of graph partitioning problems whose two extreme representatives are the well-known Min Cut and Graph Bisection problems. The former is known to be efficiently solvable by flow techniques, the latter to be NP-complete. The results presented in this paper are n n na monotony result of the type“ The more balanced the partition we look for has to be, the harder the problem”. n n na complexity result clarifying the status of a large part of intermediate problems in the class.

Modeling hypergraphs by graphs with the same mincut properties

Abstract An elegant and general way to apply graph partitioning algorithms to hypergraphs would be to model hypergraphs by graphs and apply the graph algorithms to these models. Of course such models have to simulate the given hypergraphs with respect to their cut properties. An edge-weighted graph (V, E) is a cut-model for an edge-weighted hypergraph (V, H) if the weight of the edges cut by any bipartition of V in the graph is the same as the weight of the hyperedges cut by the same bipartition in the hypergraph. We show that there is no cut-model in general. Next we examine whether the addition of dummy vertices helps: An edge-weighted graph (V ∪ D, E) is a mincut-model for an edge-weighted hypergraph (V, H) if the weight of the hyperedges cut by a bipartition of the hypergraphs vertices is the same as the weight of a minimum cut separating the two parts in the graph. We construct such models using positive and negative weights. On the other hand, we show that there is no mincut-model in general if only positive weights are allowed.

Three Rules Suffice for Good Label Placement

Abstract. The general label-placement problem consists in labeling a set of features (points, lines, regions) given a set of candidates (rectangles, circles, ellipses, irregularly shaped labels) for each feature. The problem arises when annotating classical cartographical maps, diagrams, or graph drawings. The size of a labeling is the number of features that receive pairwise nonintersecting candidates. Finding an optimal solution, i.e., a labeling of maximum size, is NP-hard. We present an approach to attack the problem in its full generality. The key idea is to separate the geometric part from the combinatorial part of the problem. The latter is captured by the conflict graph of the candidates. We present a set of rules that simplify the conflict graph without reducing the size of an optimal solution. Combining the application of these rules with a simple heuristic yields near-optimal solutions. We study competing algorithms and do a thorough empirical comparison on point-labeling data. The new algorithm we suggest is fast, simple, and effective.

Map labeling heuristics: provably good and practically useful

The lettering of maps ts a classical problem of cartography that conststs of placing names, symbols, or other data near to specified sites on a map. Certain design rules have to be obeyed. A practically interesting spe ctal case, the Map Labeling Problem, consists of placing azzs parallel rectangular labels of common size so that one of its corners w the szte, no two labels overlap, and the labels are of mazzmum size in order to have legible inscriptions. The problem w NP-hard; tt as even AfP-hard to approximate the solution with quality guaranty better than 50 percent. There w an approximation algorithm A with a qualzty guaranty of 50 percent and running ttme Q(n log n). So A M the best possible algortthm from a theoretical point of vzew. This is even true for the running tzme, stnce there M a lower bound on the running tame of any such approzimatton algorithm of Q(n log n). Unfortunately A M useless in practtce as d typically produces resuits that are Intolerably far off the maximum size. The mum contribution of this paper as the presentation of a heuristtcal approach that has A‘s advantages whtle avoiding tts disadvantages: 1. It uses A‘s result in order to guaranty the same opti*This work was done at the Institut fiir Informatik, Fachbereich Mathematilc und Informatik, Freie tJniversitiit Berlin, Takustrafie 9, 14195 BerlinDahlem, Germany. It was snpported by the ESPRIT BRA Project ALCOM II. t wagner@math .fu-berlin.de : awolff@inf .fu-berlin .de Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission’ of the Association of Computing Machinery.To copy otherwise, or to republish, requires a fee and/or specific permission. 1lth Computational Geometry, Vancouver, B.C. Canada G 1995 ACM 0-89791 -724 -3/9510006 ...

Locating New Stops in a Railway Network

3.50 mal running time eficiency; a method wh~eh M new as far as we know. 2. Its practtcal results are close to the opttmum. The practical quality M analysed by comparing our results to the exact opttmum, where thts is known; and to lower and upper bounds on the opttmum otherwise. The sample data consists of three different classes of random problems and a selection of problems artstng an the production of groundwater quality maps by the authorities of the City of Mtinchen.

A Simple Min Cut Algorithm

Abstract Given a railway network together with information on the population and their use of the railway infrastructure, we are considering the effects of introducing new train stops in the existing railway network. One effect concerns the accessibility of the railway infrastructure to the population, measured in how far people live from their nearest train stop. The second effect we study is the change in travel time for the railway customers that is induced by new train stops. Based on these two models, we introduce two combinatorial optimization problems and give NP-hardness results for them. We suggest an algorithmic approach for the model based on travel time and present a real-world application with its first experimental results.

A practical map labeling algorithm

We present an algorithm for finding the minimum cut of an edge-weighted graph. It is simple in every respect. It has a short and compact description, is easy to implement and has a surprisingly simple proof of correctness. Its runtime matches that of the fastest algorithm known. The runtime analysis is straightforward. In contrast to nearly all approaches so far, the algorithm uses no flow techniques. Roughly speaking the algorithm consists of about ¦V¦ nearly identical phases each of which is formally similar to Prims minimum spanning tree algorithm.

Fingolimod (FTY720) in Severe Hepatic Impairment: Pharmacokinetics and Relationship to Markers of Liver Function

Abstract The map labeling problem is a classical problem of cartography. There is a theoretically optimal approximation algorithm A . Unfortunately A is useless in practice as it typically produces results that are intolerably far off the optimal size. On the other hand there are heuristics with good practical results. In this paper we present an algorithm B that (1) guarantees the optimal approximation quality and runtime behaviour of A , and (2) yields results significantly closer to the optimum than the best heuristic known so far. The sample data used in the experimental evaluation consists of three different classes of random problems and a selection of problems arising in the production of groundwater quality maps by the authorities of the City of Munich.