We introduce a network creation game in which each player (vertex) has a fixed budget to establish links to other players. In this model, each link has a unit price, and each agent tries to minimize its cost, which is either its eccentricity or its total distance to other players in the underlying (undirected) graph of the created network. Two versions of the game are studied: In the MAX version, the cost incurred to a vertex is the maximum distance between the vertex and other vertices, and, in the SUM version, the cost incurred to a vertex is the sum of distances between the vertex and other vertices. We prove that in both versions pure Nash equilibria exist, but the problem of finding the best response of a vertex is NP-hard. We take the social cost of the created network to be its diameter, and next we study the maximum possible diameter of an equilibrium graph with n vertices in various cases. When the sum of players’ budgets is n − 1, the equilibrium graphs are always trees, and we prove that their maximum diameter is Θ( n ) and Θ(log n ) in MAX and SUM versions, respectively. When each vertex has a unit budget (i.e., can establish a link to just one vertex), the diameter of any equilibrium graph in either version is Θ(1). We give examples of equilibrium graphs in the MAX version, such that all vertices have positive budgets and yet the diameter is Ω(√log n ). This interesting (and perhaps counterintuitive) result shows that increasing the budgets may increase the diameter of equilibrium graphs and hence deteriorate the network structure. Then we prove that every equilibrium graph in the SUM version has diameter 2 O (√log n ) . Finally, we show that if the budget of each player is at least k , then every equilibrium graph in the SUM version is k -connected or has a diameter smaller than 4.

In the node-weighted prize-collecting Steiner tree problem (NW-PCST) we are given an undirected graph G = (V, E), non-negative costs c(u) and penalties π(u) for each u ∈ V . The goal is to find a tree T that minimizes the total cost of the vertices spanned by T plus the total penalty of vertices not in T. This problem is well-known to be set-cover hard to approximate. Moss and Rabani (STOC'01) presented a primal-dual Lagrangean-multiplier-preserving O(ln |V |)-approximation algorithm for this problem. We show a serious problem with the algorithm, and present a new, fundamentally different primal-dual method achieving the same performance guarantee. Our algorithm introduces several novel features to the primal-dual method that may be of independent interest.

We consider a network creation game in which, each player (vertex) has a limited budget to establish links to other players. In our model, each link has a unit cost and each agent tries to minimize its cost which is its local diameter or its total distance to other players in the (undirected) underlying graph of the created network. Two variants of the game are studied: in the MAX version, the cost incurred to a vertex is the maximum distance between that vertex and other vertices, and in the SUM version, the cost incurred to a vertex is the sum of distances between that vertex and other vertices. We prove that in both versions pure Nash equilibria exist, but the problem of finding the best response of a vertex is NP-hard.