  We study the complexity of several combinatorial problems in the model of
binary networked public goods games. In this game, players are represented by
vertices in a network, and the action of each player can be either investing or
not investing in a public good. The payoff of each player is determined by the
number of neighbors who invest and her own action. We study the complexity of
computing action profiles that are Nash equilibrium, and those that provide the
maximum utilitarian or egalitarian social welfare. We show that these problems
are generally NP-hard but become polynomial-time solvable when the given
network is restricted to some special domains, including networks with a
constant bounded treewidth, and those whose critical clique graphs are forests.
