We study the existence and computational complexity of coalitional stability concepts based on social networks. Our concepts represent a natural and rich combinatorial generalization of a recent notion termed partition equilibrium [5]. We assume that players in a strategic game are embedded in a social (or, communication) network, and there are coordination constraints defining the set of coalitions that can jointly deviate in the game. A main feature of our approach is that players act in a fashion to ignore potentially profitable (group) deviations if the change in their strategy may cause a decrease of utility to their neighbors in the network. We explore the properties of such considerate equilibria in application to the celebrated class of resource selection games (RSGs). Our main result proves existence of a super-strong considerate equilibrium in all symmetric RSGs with strictly increasing delays, for any social network among the players and feasible coalitions represented by the set of cliques. The existence proof is constructive and yields an efficient algorithm. In fact, the computed considerate equilibrium is a Nash equilibrium for a standard RSG, thus showing that there exists a state that is stable against selfish and considerate behavior simultaneously. Furthermore, we provide results on convergence of considerate dynamics.

We define a new class of games, congestion games with load-dependent failures (CGLFs), which generalizes the well-known class of congestion games, by incorporating the issue of resource failures into congestion games. In a CGLF, agents share a common set of resources, where each resource has a cost and a probability of failure. Each agent chooses a subset of the resources for the execution of his task, in order to maximize his own utility. The utility of an agent is the difference between his benefit from successful task completion and the sum of the costs over the resources he uses. CGLFs possess two novel features. It is the first model to incorporate failures into congestion settings, which results in a strict generalization of congestion games. In addition, it is the first model to consider load-dependent failures in such framework, where the failure probability of each resource depends on the number of agents selecting this resource. Although, as we show, CGLFs do not admit a potential function, and in general do not have a pure strategy Nash equilibrium, our main theorem proves the existence of a pure strategy Nash equilibrium in every CGLF with identical resources and nondecreasing cost functions.

Abstract In this article we study the group Steiner network problem, which is defined in the following way. Given a graph G = (V,E), a partition of its vertices into K groups and connectivity requirements between the different groups, the aim is to find simultaneously a set of representatives, one for each group, and a minimum cost connected subgraph that satisfies the connectivity requirements between the groups (representatives). This problem is a generalization of the Steiner network problem and the group Steiner tree problem, two known NP‐complete problems. We present an approximation algorithm for a special case of the group Steiner network problem with an approximation ratio of min {2(1 + 2x),2I}, where I is the cardinality of the largest group and x is a parameter that depends on the cost function. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 49(2), 160–167 2007

We introduce a new class of games, congestion games with failures (CGFs), which extends the class of congestion games to allow for facility failures. In a basic CGF (BCGF) agents share a common set of facilities (service providers), where each service provider (SP) may fail with some known probability. For reliability reasons, an agent may choose a subset of the SPs in order to try and perform his task. The cost of an agent for utilizing any SP is a function of the total number of agents using this SP. A main feature of this setting is that the cost for an agent for successful completion of his task is the minimum of the costs of his successful attempts. We show that although BCGFs do not admit a potential function, and thus are not isomorphic to classic congestion games, they always possess a pure-strategy Nash equilibrium. We also show that the SPs' congestion experienced in different Nash equilibria is (almost) unique. For the subclass of symmetric BCGFs we give a characterization of best and worst Nash equilibria. We extend the basic model by making task submission costly and define a model for taxed CGFs (TCGFs). We prove the existence of a pure-strategy Nash equilibrium for quasi-symmetric TCGFs, and present an efficient algorithm for constructing such Nash equilibrium in symmetric TCGFs.

Several problems, including the maximum integral two-flow problem, are known to be NP-complete, but efficiently solvable for planar graphs. In this paper, we extend the algorithm for maximum integral two-flow in planar graphs to certain undirected K3,3-free graphs (graphs not containing any subgraph homeomorphic to K3,3). For such graphs, we provide an O(n) algorithm for determining a maximum integral two-flow of a value not less than the value of a maximum two-flow minus two. © 1998 John Wiley & Sons, Inc. Networks 32: 67–76, 1998

We consider mixed networks, which may include both directed and undirected edges. For a nontrivial vertex subset S, an S-disconnecting set is a set of edges and vertices which intersects every path from any vertex in S to any vertex not in S. Given nonnegative edge and vertex costs, we show that the minimum cost of an S-disconnecting set defines a submodular function. This implies that the set of all S inducing minimum-cost disconnecting sets is the set of closures of a binary relation, thus extending Picard-Queyranne's (1980) result on ordinary minimum cuts. We apply this result to two-pair multicommodity problems in undirected networks, extending Hu's (1963) result to disconnecting sets that may include vertices as well as edges. These results and a result of Provan and Shier (1994) may be used for generating all sets S that induce such minimum-cost disconnecting sets and ranking such sets in order of corresponding costs, for both one-pair problems in mixed networks and two-pair problems in undirected networks. © 1996 John Wiley & Sons, Inc.