The Communication Complexity of Combinatorial Auctions with Additional Succinct Bidders

We study the communication complexity of welfare maximization in combinatorial auctions with bidders from either a standard valuation class (which require exponential communication to state, such as subadditive or XOS), or arbitrary succinct valuations (which can be fully described in polynomial communication, such as single-minded). Although succinct valuations can be efficiently communicated, we show that additional succinct bidders have a nontrivial impact on communication complexity of classical combinatorial auctions. Specifically, let n be the number of subadditive/XOS bidders. We show that for SA ∪ Succ (the union of subadditive and succinct valuations): (1) there is a polynomial communication 3-approximation algorithm; (2) as n → ∞, there is a matching 3-hardness of approximation, which is larger than the optimal approximation ratio of 2 for SA, and holds even for SA ∪ SM (the union of subadditive and single-minded valuations); (3) for all n ≥ 3, there is a constant separation between the optimal approximation ratios for SA ∪ SM and SA (and therefore between SA ∪ Succ and SA as well). Similarly, we show that for XOS ∪ Succ: (1) there is a polynomial communication 2-approximation algorithm; (2) as n → ∞, there is a matching 2-hardness of approximation, which is larger than the optimal approximation ratio of e/(e−1) for XOS, and holds even for XOS ∪ SM; (3) for all n ≥ 2, there is a constant separation between the optimal approximation ratios for XOS ∪ SM and XOS (and therefore between XOS ∪ Succ and XOS as well).