Thomas Erlebach

Constant-factor approximation for minimum-weight (connected) dominating sets in unit disk graphs

For a given graph with weighted vertices, the goal of the minimum-weight dominating set problem is to compute a vertex subset of smallest weight such that each vertex of the graph is contained in the subset or has a neighbor in the subset. A unit disk graph is a graph in which each vertex corresponds to a unit disk in the plane and two vertices are adjacent if and only if their disks have a non-empty intersection. We present the first constant-factor approximation algorithm for the minimum-weight dominating set problem in unit disk graphs, a problem motivated by applications in wireless ad-hoc networks. The algorithm is obtained in two steps: First, the problem is reduced to the problem of covering a set of points located in a small square using a minimum-weight set of unit disks. Then, a constant-factor approximation algorithm for the latter problem is obtained using enumeration and dynamic programming techniques exploiting the geometry of unit disks. Furthermore, we also show how to obtain a constant-factor approximation algorithm for the minimum-weight connected dominating set problem in unit disk graphs.

Optimal bandwidth reservation in hose-model VPNs with multi-path routing

A virtual private network (VPN) provides private network connections over a publicly accessible shared network. Bandwidth provisioning for VPNs leads to challenging optimization problems. In the hose model proposed by Duffield et al., each VPN endpoint specifies bounds on the total amount of traffic that it will send or receive at any time. The network provider must provision the VPN so that there is sufficient bandwidth for any traffic matrix that is consistent with these bounds. While previous work has considered tree routing and single-path routing between the VPN endpoints, we demonstrate that the use of multipath routing offers significant advantages. On the one band, we present an optimal polynomial-time algorithm that computes a bandwidth reservation of minimum cost using multi-path routing. This is in contrast to tree routing and single-path routing, where the problem is computationally hard. On the other hand, we present experimental results showing that the reservation cost using multi-path routing can indeed be significantly smaller than with tree or single-path routing.

The complexity of path coloring and call scheduling

Abstract Modern high-performance communication networks pose a number of challenging problems concerning the efficient allocation of resources to connection requests. In all-optical networks with wavelength-division multiplexing, connection requests must be assigned paths and colors (wavelengths) such that intersecting paths receive different colors, and the goal is to minimize the number of colors used. This path coloring problem is proved NP -hard for undirected and bidirected ring networks. Path coloring in undirected tree networks is shown to be equivalent to edge coloring of multigraphs, which implies a polynomial-time optimal algorithm for trees of constant degree as well as NP -hardness and an approximation algorithm with absolute approximation ratio 4 3 and asymptotic approximation ratio 1.1 for trees of arbitrary degree. For bidirected trees, path coloring is shown to be NP -hard even in the binary case. A polynomial-time optimal algorithm is given for path coloring in undirected or bidirected trees with n nodes under the assumption that the number of paths touching every single node of the tree is O (( log n) 1−e ) . Call scheduling is the problem of assigning paths and starting times to calls in a network with bandwidth reservation such that the maximum completion time is minimized. In the case of unit bandwidth requirements, unit edge capacities, and unit call durations, call scheduling is equivalent to path coloring. If either the bandwidth requirements or the call durations can be arbitrary, call scheduling is shown NP -hard for virtually every network topology.

Computing the types of the relationships between autonomous systems

We investigate the problem of computing the types of the relationships between Internet Autonomous Systems. We refer to the model introduced by Gao [IEEE/ACM Transactions on Networking, 9(6):733-645, 2001] and Subramanian (IEEE Infocom, 2002) that bases the discovery of such relationships on the analysis of the AS paths extracted from the BGP routing tables. We characterize the time complexity of the above problem, showing both NP-completeness results and efficient algorithms for solving specific cases. Motivated by the hardness of the general problem, we propose approximation algorithms and heuristics based on a novel paradigm and show their effectiveness against publicly available data sets. The experiments provide evidence that our algorithms perform significantly better than state-of-the-art heuristics

Call scheduling in trees, rings and meshes

The problem of establishing and completing a given set of calls as early as possible is studied for bidirectional and directed calls in various classes of networks. Even under the assumption of unit bandwidth requirements and unit call durations, call scheduling is NP-hard for trees with unbounded degree, for rings, and for meshes. Whereas bidirectional calls can be scheduled optimally in polynomial time for trees of constant degree, the problem for directed calls is already NP-hard for binary trees. Approximation algorithms with a constant performance ratio are known for many NP-hard variants of call scheduling.

On the Spectrum and Structure of Internet Topology Graphs

In this paper we study properties of the Internet topology on the autonomous system (AS) level. We find that the normalized Laplacian spectrum (nls) of a graphpro vides a concise fingerprint of the corresponding network topology. The nls of AS graphs remains stable over time in spite of the explosive growth of the Internet, but the nls of synthetic graphs obtained using the state-of-the-art topology generator Inet-2.1 is significantly different, in particular concerning the multiplicity of eigenvalue 1. We relate this multiplicity to the sizes of certain subgraphs and thus obtain a new structural classification of the nodes in the AS graphs, which is also plausible in networking terms. These findings as well as new power-law relationships discovered in the interconnection structure of the subgraphs may lead to a new generator that creates more realistic topologies by combining structural and power-law properties.

A (4+ε)-approximation for the minimum-weight dominating set problem in unit disk graphs

We present a (4+e)-approximation algorithm for the problem of computing a minimum-weight dominating set in unit disk graphs, where e is an arbitrarily small constant. The previous best known approximation ratio was 5+e. The main result of this paper is a 4-approximation algorithm for the problem restricted to constant-size areas. To obtain the (4+e)-approximation algorithm for the unrestricted problem, we then follow the general framework from previous constant-factor approximations for the problem: We consider the problem in constant-size areas, and combine the solutions obtained by our 4-approximation algorithm for the restricted case to get a feasible solution for the whole problem. Using the shifting technique (selecting a best solution from several considered partitionings of the problem into constant-size areas) we obtain the claimed (4+e)-approximation algorithm. By combining our algorithm with a known algorithm for node-weighted Steiner trees, we obtain a 7.875-approximation for the minimum-weight connected dominating set problem in unit disk graphs.

Constrained Bipartite Edge Coloring with Applications to Wavelength Routing

Motivated by the problem of efficient routing in all-optical networks, we study a constrained version of the bipartite edge coloring problem. We show that if the edges adjacent to a pair of opposite vertices of an L-regular bipartite graph are already colored with αL different colors, then the rest of the edges can be colored using at most (1+α/2)L colors. We also show that this bound is tight by constructing instances in which (1+α/2)L colors are indeed necessary. We also obtain tight bounds on the number of colors that each pair of opposite vertices can see.

The Maximum Edge-Disjoint Paths Problem in Bidirected Trees

A bidirected tree is the directed graph obtained from an undirected tree by replacing each undirected edge by two directed edges with opposite directions. Given a set of directed paths in a bidirected tree, the goal of the maximum edge-disjoint paths problem is to select a maximum-cardinality subset of the paths such that the selected paths are edge-disjoint. This problem can be solved optimally in polynomial time for bidirected trees of constant degree but is APX-hard for bidirected trees of arbitrary degree. For every fixed

Learning One-Variable Pattern Languages Very Efficiently on Average, in Parallel, and by Asking Queries

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