Vickrey-Clarke-Groves (VCG) mechanisms are a well-known framework for finding a solution to a distributed optimization problem in systems of self-interested agents. VCG mechanisms have received wide attention in the AI community because they are efficient and strategy-proof; a special case of the Groves family of mechanisms, VCG mechanisms are the only direct-revelation mechanisms that are allocatively efficient and strategy-proof. Unfortunately, VCG mechanisms are only weakly budget-balanced. We consider self-interested agents in a network flow domain, and show that in this domain, it is possible to design a mechanism that is both allocatively-efficient and almost completely budget-balanced. This is done by choosing a mechanism that is not strategy-proof but rather strategy-resistant. Instead of using the VCG mechanism, we propose a mechanism in which finding the most beneficial manipulation is an NP-complete problem, and the payments from the agents to the mechanism may be minimized as much as desired. This way, the mechanism is virtually strongly budget-balanced: for any ε > 0, we find a mechanism that is ε-budget-balanced.