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 explicitly 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 $\cup$ SC (the union of subadditive and succinct valuations): (1) There is a polynomial communication $3$-approximation algorithm; (2) As $n \to \infty$, there is a matching $3$-hardness of approximation, which (a) is larger than the optimal approximation ratio of $2$ for SA, and (b) holds even for SA $\cup$ SM (the union of subadditive and single-minded valuations); and (3) For all $n \geq 3$, there is a constant separation between the optimal approximation ratios for SA $\cup$ SM and SA (and therefore between SA $\cup$ SC and SA as well). Similarly, we show that for XOS $\cup$ SC: (1) There is a polynomial communication $2$-approximation algorithm; (2) As $n \to \infty$, there is a matching $2$-hardness of approximation, which (a) is larger than the optimal approximation ratio of $e/(e-1)$ for XOS, and (b) holds even for XOS $\cup$ SM; and (3) For all $n \geq 2$, there is a constant separation between the optimal approximation ratios for XOS $\cup$ SM and XOS (and therefore between XOS $\cup$ SC and XOS as well).