We provide approximation algorithms for several variants of the FIRE-FIGHTER problem on general graphs. The Firefighter problem models the case where a diffusive process such as an infection (or an idea, a computer virus, a fire) is spreading through a network, and our goal is to contain this infection by using targeted vaccinations. Specifically, we are allowed to vaccinate at most a fixed number (called the budget) of nodes per time step, with the goal of minimizing the effect of the infection. The difficulty of this problem comes from its temporal component, since we must choose nodes to vaccinate at every time step while the infection is spreading through the network, leading to notions of "cuts over time".We consider two versions of the Firefighter problem: a "non-spreading" model, where vaccinating a node means only that this node cannot be infected; and a "spreading" model where the vaccination itself is an infectious process, such as in the case where the infection is a harmful idea, and the vaccine to it is another infectious beneficial idea. We look at two measures: the MAXSAVE measure in which we want A preliminary version of this paper appeared in ISAAC 2009. Algorithmica (2012 to maximize the number of nodes which are not infected given a fixed budget, and the MINBUDGET measure, in which we are given a set of nodes which we have to save and the goal is to minimize the budget. We study the approximability of these problems in both models.

Abstract. We provide approximation algorithms for several variants of the Firefighter problem on general graphs. The Firefighter problem models the case where an infection or another diffusive process (such as an idea, a computer virus, or a fire) is spreading through a network, and our goal is to stop this infection by using targeted vaccinations. Specifically, we are allowed to vaccinate at most B nodes per time-step (for some budget B), with the goal of minimizing the effect of the infection. The difficulty of this problem comes from its temporal component, since we must choose nodes to vaccinate at every time-step while the infection is spreading through the network, leading to notions of "cuts over time".We consider two versions of the Firefighter problem: a "non-spreading" model, where vaccinating a node means only that this node cannot be infected; and a "spreading" model where the vaccination itself is an infectious process, such as in the case where the infection is a harmful idea, and the vaccine to it is another infectious idea. We give complexity and approximation results for problems on both models.

We consider the existence of Partition Equilibrium in Resource Selection Games. Super-strong equilibrium, where no subset of players has an incentive to change their strategies collectively, does not always exist in such games. We show, however, that partition equilibrium (introduced in [4] to model coalitions arising in a social context) always exists in general resource selection games, as well as how to compute it efficiently. In a partition equilibrium, the set of players has a fixed partition into coalitions, and the only deviations considered are by coalitions that are sets in this partition. Our algorithm to compute a partition equilibrium in any resource selection game (i.e., load balancing game) settles the open question from [4] about existence of partition equilibrium in general resource selection games. Moreover, we show how to always find a partition equilibrium which is also a Nash equilibrium. This implies that in resource selection games, we do not need to sacrifice the stability of individual players when forming solutions stable against coalitional deviations. In addition, while super-strong equilibrium may not exist in resource selection games, we show that its existence can be decided efficiently, and how to find one if it exists.

Abstract. We consider the existence of Partition Equilibrium in Resource Selection Games. Super-strong equilibrium, where no subset of players has an incentive to change their strategies collectively, does not always exist in such games. We show, however, that partition equilibrium (introduced in [4] to model coalitions arising in a social context) always exists in general resource selection games, as well as how to compute it efficiently. In a partition equilibrium, the set of players has a fixed partition into coalitions, and the only deviations considered are by coalitions that are sets in this partition. Our algorithm to compute a partition equilibrium in any resource selection game (i.e., load balancing game) settles the open question from [4] about existence of partition equilibrium in general resource selection games. Moreover, we show how to always find a partition equilibrium which is also a Nash equilibrium. This implies that in resource selection games, we do not need to sacrifice the stability of individual players when forming solutions stable against coalitional deviations. In addition, while super-strong equilibrium may not exist in resource selection games, we show that its existence can be decided efficiently, and how to find one if it exists.

We consider a model of next-hop routing by self-interested agents. In this model, nodes in a graph (representing ISPs, Autonomous Systems, etc.) make pricing decisions of how much to charge for forwarding traffic from each of their upstream neighbors, and routing decisions of which downstream neighbors to forward traffic to (i.e., choosing the next hop). Traffic originates at a subset of these nodes that derive a utility when the traffic is routed to its destination node; the traffic demand is elastic and the utility derived from it can be different for different source nodes. Our next-hop routing and pricing model is in sharp contrast with the more common source routing and pricing models, in which the source of traffic determines the entire route from source to destination. For our model, we begin by showing sufficient conditions for prices to result in a Nash equilibrium, and in fact give an efficient algorithm to compute a Nash equilibrium which is as good as the centralized optimum, thus proving that the price of stability is 1. When only a single source node exists, then the price of anarchy is 1 as well, as long as some minor assumptions on player behavior is made. The above results hold for arbitrary convex pricing functions, but with the assumption that the utilities derived from getting traffic to its destination are linear. When utilities can be non-linear functions, we show that Nash equilibrium may not exist, even with simple discrete pricing models.