The Communication Complexity of Combinatorial Auctions with Additional Succinct Bidders

TL;DR

This paper investigates how additional succinct bidders influence the communication complexity in combinatorial auctions, revealing new approximation algorithms and hardness results that differ from classical settings.

Contribution

It provides the first analysis of communication complexity with succinct bidders, establishing polynomial algorithms and matching hardness bounds for welfare maximization.

Findings

Polynomial 3-approximation for SA ∪ SC with increasing n

Matching hardness of approximation larger than classical bounds

Constant separation between approximation ratios for SA ∪ SM and SA

Abstract

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).