A combinatorial auction is an auction, in which the price and allocation of multiple goods are decided simultatiously within an instant. In the setting of graphical valuations, each bidder tells the auctioneer beforehands how much they would be willing to pay for each subset of goods. This valuation is controlled by a graph, for which weights on the vertices and edges define the valuation for each subset of goods. Given these valuations, the auctioneer decides a price per item and a distribution of the goods among the bidders.
A bidder is happy with the outcome of an aution, if they receive a subset of goods at a price, which maximizes their surplus. We pose the question of the existence of a price and an allocation, at which all bidders are happy; this is a competitive equilibrium.