text: Network Creation Games are a well-known approach for explaining and analyzing the structure, quality and dynamics of real-world networks like the Internet and other infrastructure networks which evolved via the interaction of selfish agents without a central authority. In these games selfish agents which correspond to nodes in a network strategically buy incident edges to improve their centrality. However, past research on these games has only considered the creation of networks with unit-weight edges. In practice, e.g. when constructing a fiber-optic network, the choice of which nodes to connect and also the induced price for a link crucially depends on the distance between the involved nodes and such settings can be modeled via edge-weighted graphs. We incorporate arbitrary edge weights by generalizing the well-known model by [PODC'03] to edge-weighted host graphs and focus on the geometric setting where the weights are induced by the distances in some metric space. In stark contrast to the state-of-the-art for the unit-weight version, where the Price of Anarchy is conjectured to be constant and where resolving this is a major open problem, we prove a tight non-constant bound on the Price of Anarchy for the metric version and a slightly weaker upper bound for the non-metric case. Moreover, we analyze the existence of equilibria, the computational hardness and the game dynamics for several natural metrics. The model we propose can be seen as the game-theoretic analogue of a variant of the classical Network Design Problem. Thus, low-cost equilibria of our game correspond to decentralized and stable approximations of the optimum network design.
  @cite_21: In the network design problem we are given a weighted undirected graph. We wish to find a subgraph which connects all the original vertices and minimizes the sum of the shortest path weights between all vertex pairs, subject to a budget constraint on the sum of its edge weights. In this note we establish NP-completeness for the network design problem, even for the simple case where all edge weights are equal and the budget restricts the choice to spanning trees. This result justifies the development of enumerative optimization methods and of approximation algorithms, such as those described in a recent paper by R. Dionne and M. Florian.
  @cite_1: Numerous transportation applications as diverse as capital investment decision-making, vehicle fleet planning, and traffic light signal setting all involve some form of (discrete choice) network design. In this paper, we review some of the uses and limitations of integer programming-based approaches to network design, and describe several discrete and continuous choice models and algorithms. Our objectives are threefold---to provide a unifying view for synthesizing many network design models, to propose a unifying framework for deriving many network design algorithms, and to summarize computational experience in solving design problems. We also show that many of the most celebrated combinatorial problems that arise in transportation planning are specializations and variations of a generic design model. Consequently, the network design concepts described in this paper have great potential application in a wide range of problem settings.
  @cite_39: 
summary: There are many classical optimization problems related to network design, e.g. see the survey by Magnanti & Wong @cite_1 . Many of them are NP-complete, e.g. all the problems labeled ND'' in @cite_39 . Our model is closely related to the Network Design Problem @cite_21 and the Optimum Communication Spanning Tree Problem (ND7 in @cite_39 ). In particular, finding the social optimum network corresponds to a variant of the Network Design Problem, where, instead of having separate budgets for buying edges and for the routing cost, the sum of edge costs and routing costs, i.e., the total distance between all pairs of nodes, is to be minimized. Hence we strongly suspect that computing the social optimum in all versions of our model, with the and the as exceptions, is NP-hard.