This paper develops a, new combinatorial auction protocol called the Groves Mechanism with SubModular Approximation (GM-SMA). This protocol satisfies the following characteristics: (1) it is false-name-proof, (2) each winner is included in a Pareto efficient allocation, and (3) as long as a Pareto efficient allocation is achieved, the protocol is robust against the collusion of losers and the outcome is in the core. As far as the authors are aware, the GM-SMA is the first protocol that satisfies all three of these characteristics. The basic ideas of the GM-SMA are as follows: (i) It is based on the VCG protocol, i.e., the payment of a winner in this protocol is identical to the payment in one instance of the Groves mechanism, which is a class of protocols that includes the VCG. (ii) When calculating the payment of a, bidder, we approximate the valuations of other bidders by using a submodular valuation function (submodular approximation). Simulation results show that the GM-SMA achieves a better social surplus and seller's revenue than existing false-name-proof protocols, as long as the submodular approximation is close enough to the original valuations.