Joachim Spoerhase

(r,p)-centroid problems on paths and trees

An instance of the (r,p)-centroid problem is given by an edge and node weighted graph. Two competitors, the leader and the follower, are allowed to place p and r facilities, respectively, into the graph. Users at the nodes connect to the closest facility. A solution of the (r,p)-centroid problem is a leader placement such that the maximum total weight of the users connecting to any follower placement is as small as possible. We show that the absolute (r,p)-centroid problem is NP-hard even on a path which answers a long-standing open question of the complexity of the problem on trees (Hakimi, 1990 [10]). Moreover, we provide polynomial time algorithms for the discrete (r,p)-centroid on paths and the (1,p)-centroid on trees, and complementary hardness results for more complex graph classes.

Multiple voting location and single voting location on trees

Abstract We examine voting location problems in which the goal is to place, based on an election amongst the users, a given number of facilities in a graph. The user preference is modeled by shortest path distances in the graph. A Condorcet solution is a set of facilities to which there does not exist an alternative set preferred by a majority of the users. Recent works generalize the model to additive indifference and replaced user majority by γ -proportion. We show that for multiple voting location, Condorcet and Simpson decision problems are Σ 2 p -complete, and investigate the approximability of the Simpson and the Simpson score optimization problem. Further we contribute a result towards lower bounds on the complexity of the single voting location problem. On the positive side we develop algorithms for the optimization problems on tree networks which are substantially faster than the existing algorithms for general graphs. Finally we suggest a generalization of the indifference notion to threshold functions.

Algorithms for Labeling Focus Regions

In this paper, we investigate the problem of labeling point sites in focus regions of maps or diagrams. This problem occurs, for example, when the user of a mapping service wants to see the names of restaurants or other POIs in a crowded downtown area but keep the overview over a larger area. Our approach is to place the labels at the boundary of the focus region and connect each site with its label by a linear connection, which is called a leader. In this way, we move labels from the focus region to the less valuable context region surrounding it. In order to make the leader layout well readable, we present algorithms that rule out crossings between leaders and optimize other characteristics such as total leader length and distance between labels. This yields a new variant of the boundary labeling problem, which has been studied in the literature. Other than in traditional boundary labeling, where leaders are usually schematized polylines, we focus on leaders that are either straight-line segments or Bezier curves. Further, we present algorithms that, given the sites, find a position of the focus region that optimizes the above characteristics. We also consider a variant of the problem where we have more sites than space for labels. In this situation, we assume that the sites are prioritized by the user. Alternatively, we take a new facility-location perspective which yields a clustering of the sites. We label one representative of each cluster. If the user wishes, we apply our approach to the sites within a cluster, giving details on demand.

Better Approximation Algorithms for the Maximum Internal Spanning Tree Problem

We examine the problem of determining a spanning tree of a given graph such that the number of internal nodes is maximum. The best approximation algorithm known so far for this problem is due to Prieto and Sloper and has a ratio of 2. For graphs without pendant nodes, Salamon has lowered this factor to

Specialized Heuristics for the Controller Placement Problem in Large Scale SDN Networks

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On Monotone Drawings of Trees

by means of local search. However, the approximative behaviour of his algorithm on general graphs has remained open. In this paper we show that a simplified and faster version of Salamons algorithm yields a

Maximum betweenness centrality: approximability and tractable cases

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Drawing graphs with vertices at specified positions and crossings at large angles

-approximation even on general graphs. In addition to this, we investigate a node weighted variant of the problem for which Salamon achieved a ratio of 2·Δ(G ) *** 3. Modifying Salamons approach we obtain a factor of 3 + *** for any *** > 0. We complement our results with worst case instances showing that our bounds are tight.

Optimally computing all solutions of Stackelberg with parametric prices and of general monotonous gain functions on a tree

The Software Defined Networking (SDN) concept introduces a paradigm shift in the networking world towards an externalized control plane which is logically centralized. When designing an SDN-based WAN architecture, it is of vital importance to find a feasible solution to the controller placement problem, i.e., to decide where to position a limited amount of resources within the network. In addition to time-independent constraints regarding aspects like scalability, resilience, and control plane communication delays, dynamically changing network conditions like traffic patterns or bandwidth demands need to be considered as well. Consequently, such dynamic environments call for a regular and fast recalculation of placements in order to adapt to the current situation in a timely manner. While an exhaustive evaluation of all possible solutions can be performed within a practically feasible time frame for small and medium-sized networks, such an approach is out of scope for large problem instances which have significantly higher time and memory requirements. Therefore, this work investigates a specialized heuristic, which takes into account a particular set of optimization objectives and returns solutions representing the possible trade-offs between them. Due to its low computation time and acceptable margin of error, this heuristic can be employed by automatic decision systems operating in dynamic environments.

Approximating minimum manhattan networks in higher dimensions

A crossing-free straight-line drawing of a graph is monotone if there is a monotone path between any pair of vertices with respect to some direction. We show how to construct a monotone drawing of a tree with n vertices on an On 1.5 ×On 1.5 grid whose angles are close to the best possible angular resolution. Our drawings are convex, that is, if every edge to a leaf is substituted by a ray, the unbounded faces form convex regions. It is known that convex drawings are monotone and, in the case of trees, also crossing-free. A monotone drawing is strongly monotone if, for every pair of vertices, the direction that witnesses the monotonicity comes from the vector that connects the two vertices. We show that every tree admits a strongly monotone drawing. For biconnected outerplanar graphs, this is easy to see. On the other hand, we present a simply-connected graph that does not have a strongly monotone drawing in any embedding.