We investigate price-based mechanisms with connectedness in combinatorial
auctions, where with restrictions of privacy and complexity, the auctioneer asks a limited number of prices to buyers who provide demand responses. Consistent with the price-based property, several necessary and sufficient conditions are presented for the existence of the VCG mechanism, strategy-proofness with participation constraints, approximate strategy-proofness, Nash equilibrium, efficiency, core, and others. In all cases, the concept of the representative valuation function, which assigns the minimal
valuation in both absolute and relative terms to any revealed package, plays the central role in determining whether these conditions are satisfied.