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In this paper, we focus on the fact that even if we ignore the computational hardness of winner determination, a combinatorial auction is in many situations unlikely to produce the most efficient allocation. This is because each bidder has an exponential number of combinations to bid on, and basically only her own valuations as help when selecting combinations. This paper is based on the conjecture that for a large class of auction instances, given a reasonable restriction on the input each bidder can provide, there exists an auction mechanism where the winner determination problem can be solved in polynomial time, at the same time as the expected result is more efficient than that of a conventional combinatorial auction. The evidence is given by explicitly describing such an auction mechanism, and by simulation results.