The Communication Complexity of Combinatorial Auctions with Additional Succinct Bidders

TL;DR

This work analyzes how adding succinct bidders to combinatorial auctions affects the communication complexity of welfare maximization for subadditive/XOS valuations. It provides poly(m,n)-bit, 3-approx algorithms for SA∪Succ and 2-approx algorithms for XOS∪Succ, while proving matching exponential-communication hardness as the number of non-succinct bidders grows. The paper further shows constant separations between SA∪SM and SA, and between XOS∪SM and XOS, and introduces strong inapproximability notions via sophisticated reductions (PromiseDisjointness, ExistsFarSets) built on information-theoretic arguments. This framework advances understanding of the trade-offs between succinct inputs and efficiency/incentive-compatibility in auction design, with implications for deterministic truthful mechanism lower bounds and poly-time welfare optimization under strategic behavior.

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).

Figures (2)

• Figure 1: An illustrative example of a hard instance for the $\mathsf{XOS}\,\cup\,\mathsf{SM}$ lower bound. Here we have $n=4$ XOS bidders, $c=6$ single-minded bidders, and $m=nc=24$ items. The valuation for $\mathsf{XOS}_i$ is defined by the collection of items $\mathcal{S}_i$ over row $i$, while $\mathsf{SM}_j$ is single-minded for the items in column $j$. The optimal allocation can only be achieved by allocating the common columns $\{5, 6\}$ to the XOS bidders and allocating the rest to the single-minded bidders. On the other hand, if the algorithm cannot identify the common column set $\{5, 6\}$, it must either fail to satisfy many single-minded bidders as in (c), or fail to satisfy many XOS bidders.
• Figure 2: The $\mathsf{SA}\,\cup\,\mathsf{SM}$ hard instance. Instead of wanting a single item from each column like in the $\mathsf{XOS}\,\cup\,\mathsf{SM}$ hard instance, each subadditive bidder has a complex valuation over the items in each column so that the optimal allocation among subadditive bidders is hard to find. It is possible for a suboptimal allocation like (c) to split the subadditive bidders across columns so that each column is "responsible" for giving value to significantly fewer subadditive bidders, since each subadditive bidder only wants an $\varepsilon$ fraction of the columns. Therefore, the standard notion of $2$-inapproximability that subadditive valuations are known to satisfy does not suffice: we need a stronger notion where almost every sub-instance of the bidders is $2$-inapproximable as well. In this example, there is only one bidder in columns $1$ and $3$, so finding an optimal allocation for it is trivial. If we could also efficiently find an optimal allocation between any two subadditive bidders, then we could recover the full subadditive welfare, meaning the instance is no harder to approximate than the simpler $\mathsf{XOS}\,\cup\,\mathsf{SM}$ hard instance. It turns out that subadditive valuations satisfy a notion of strong $2$-inapproximability that allows for the $\mathsf{SA}\,\cup\,\mathsf{SM}$ hard instance to work as intended. Note that such a property is not a given; for example, XOS valuations do not satisfy any such strong inapproximability notion, despite being $e/(e-1)$-inapproximable in the standard sense. This is why the $\mathsf{XOS}\,\cup\,\mathsf{SM}$ hard instance cannot be made meaningfully harder with more complexity within each column.

Theorems & Definitions (113)

• Theorem 1.1: name=,restate=ComplexitySASM
• Theorem 1.2: name=,restate=ComplexityXOSSM
• Theorem 1.3: name=,restate=SeparationSASM
• Theorem 1.4: name=,restate=SeparationXOSSM
• Theorem 1.5: Feige09DobzinskiNS10
• Definition 1.6: Scarce
• Definition 1.7: Strong Inapproximability
• Theorem 1.8: name=,restate=SubadditiveStronglyInapproximableThm
• Definition 2.1: $\alpha$-Approximation
• Definition 2.2: $\alpha$-Approximation