Martin Hoefer

On Modularity Clustering

Modularity is a recently introduced quality measure for graph clusterings. It has immediately received considerable attention in several disciplines, particularly in the complex systems literature, although its properties are not well understood. We study the problem of finding clusterings with maximum modularity, thus providing theoretical foundations for past and present work based on this measure. More precisely, we prove the conjectured hardness of maximizing modularity both in the general case and with the restriction to cuts and give an Integer Linear Programming formulation. This is complemented by first insights into the behavior and performance of the commonly applied greedy agglomerative approach.

On finding graph clusterings with maximum modularity

Modularity is a recently introduced quality measure for graph clusterings. It has immediately received considerable attention in several disciplines, and in particular in the complex systems literature, although its properties are not well understood. We study the problem of finding clusterings with maximum modularity, thus providing theoretical foundations for past and present work based on this measure. More precisely, we prove the conjectured hardness of maximizing modularity both in the general case and with the restriction to cuts, and give an Integer Linear Programming formulation. This is complemented by first insights into the behavior and performance of the commonly applied greedy agglomaration approach.

Approximation algorithms for secondary spectrum auctions

We study combinatorial auctions for the secondary spectrum market. In this market, short-term licenses shall be given to wireless nodes for communication in their local neighborhood. In contrast to the primary market, channels can be assigned to multiple bidders, provided that the corresponding devices are well separated such that the interference is sufficiently low. Interference conflicts are described in terms of a conflict graph in which the nodes represent the bidders and the edges represent conflicts such that the feasible allocations for a channel correspond to the independent sets in the conflict graph. In this paper, we suggest a novel LP formulation for combinatorial auctions with conflict graph using a non-standard graph parameter, the so-called inductive independence number. Taking into account this parameter enables us to bypass the well-known lower bound of Ω(n1-ε) on the approximability of independent set in general graphs with n nodes (bidders). We achieve significantly better approximation results by showing that interference constraints for wireless networks yield conflict graphs with bounded inductive independence number. Our framework covers various established models of wireless communication, e.g., the protocol or the physical model. For the protocol model, we achieve an O(√k)-approximation, where k is the number of available channels. For the more realistic physical model, we achieve an O(√k log2 n) approximation based on edge-weighted conflict graphs. Combining our approach with the LP-based framework of Lavi and Swamy, we obtain incentive compatible mechanisms for general bidders with arbitrary valuations on bundles of channels specified in terms of demand oracles.

Stackelberg Network Pricing Games

We study a multi-player one-round game termed Stackelberg Network Pricing Game, in which a leader can set prices for a subset of

Scheduling in Wireless Networks with Rayleigh-Fading Interference

m

On the Complexity of Pareto-Optimal Nash and Strong Equilibria

priceable edges in a graph. The other edges have a fixed cost. Based on the leaders decision one or more followers optimize a polynomial-time solvable combinatorial minimization problem and choose a minimum cost solution satisfying their requirements based on the fixed costs and the leaders prices. The leader receives as revenue the total amount of prices paid by the followers for priceable edges in their solutions, and the problem is to find revenue maximizing prices. Our model extends several known pricing problems, including single-minded and unit-demand pricing, as well as Stackelberg pricing for certain follower problems like shortest path or minimum spanning tree. Our first main result is a tight analysis of a single-price algorithm for the single follower game, which provides a

Geometric Network Design with Selfish Agents

(1+varepsilon) log m

Computing pure Nash and strong equilibria in bottleneck congestion games

-approximation for any

Network Creation Games with Disconnected Equilibria

varepsilon >0

Non-cooperative facility location and covering games

. This can be extended to provide a