This paper presents a formulation and an algorithm for the winner determination problem in auction-based scheduling. Without imposing a time line discretization, the proposed approach allows bidders to bid for the processing of a set of jobs using a requirement-based bidding language, which naturally represents scheduling constraints. The proposed winner determination algorithm uses a depth first branch and bound search. The search branches on bids, and a constraint-directed scheduling procedure is used at each node to verify the feasibility of the temporary schedule. Experiments show that the proposed algorithm is on average more than an order of magnitude faster than a commercial optimization package, CPLEX 10.0.