Abstract-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 agglomerative approach.

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.

We study a multi-player one-round game termed Stackelberg Network Pricing Game, in which a leader can set prices for a subset of m priceable edges in a graph. The other edges have a fixed cost. Based on the leader's 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 leader's prices. The leader receives as revenue the total amount of prices paid by the followers for priceable edges in their solutions. 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 (1 + ε) log m-approximation. This can be extended to provide a (1 + ε)(log k + log m)-approximation for the general problem and k followers. The problem is also shown to be hard to approximate within O(log ε k + log ε m) for some ε > 0. If followers have demands, the single-price algorithm provides an O(m 2 )-approximation, and the problem is hard to approximate within O(m ) for some ε > 0. Our second main result is a polynomial time algorithm An extended abstract of this paper has appeared in STACS 2008 [11].M. Hoefer is supported by DFG Graduiertenkolleg "AlgoSyn". P. Krysta is supported by DFG grant Kr 2332/1-2 within Emmy Noether program.734 Algorithmica (2012) 62:733-753 for revenue maximization in the special case of Stackelberg bipartite vertex-cover, which is based on non-trivial max-flow and LP-duality techniques. This approach can be extended to provide constant-factor approximations for any constant number of followers.

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