This paper studies shill bidding in the VCG mechanism applied to combinatorial auctions. Shill bidding is a strategy whereby a single decision-maker enters the auction under the guise of multiple identities (Sakurai, Yokoo, and Matsubara 1999). I formulate the problem of optimal shill bidding for a bidder who knows the aggregate bid of her opponents. A key to the analysis is a subproblem---the cost minimization problem---which searches for the cheapest way to win a given package using shills. This formulation leads to an exact characterization of the aggregate bids b such that some bidder would have an incentive to shill bid against b. It is well known that when goods are substitutes, there is no incentive to shill bid. In contrast, I show that when goods are pure complements, the incentive to shill takes a simple form: there is an incentive to disintegrate and bid for each item using a dierent identity. With a mix of substitutes and complements, I show that the winner determination problem (for single minded bidders)---the problem of finding an efficient allocation in
a combinatorial auction---can be embedded into the optimal shill bidding problem. Shill bidding is closely related to collusion. Setting aside the ordinary incentive to suppress competition, the disincentive to disintegrate using shills when facing a substitutes valuation translates into an incentive to merge for a coalition facing the same valuation. Only when valuations are additive can the incentives to shill and merge simultaneously disappear. The paper also shows that there does not exist a dominant strategy in the VCG mechanism when shill bidding is possible. I find a large class of shill bidding strategies which sometimes outperform truthful bidding, but also show that no shill bidding strategy dominates truthful bidding.