Complements between goods --- where one good takes on added value in the presence of another --- have been a thorn in the side of algorithmic mechanism designers. On the one hand, complements are common in the standard motivating applications for combinatorial auctions, like spectrum license auctions. On the other, welfare maximization in the presence of complements is notoriously difficult, and this intractability has stymied theoretical progress in the area. For example, there are no known positive results for combinatorial auctions in which bidder valuations are multi-parameter and non-complement-free, other than the relatively weak results known for general valuations.

To make inroads on the problem of combinatorial auction design in the presence
of complements, we propose a model for valuations with complements that is
parameterized by the “size” of the complements. The model permits a succinct
representation, a variety of computationally efficient queries, and non-trivial
welfare-maximization algorithms and mechanisms. Specifically, a hypergraph-r
valuation v for a good set M is represented by a hypergraph H=(M,E), where every
(hyper-)edge e in E contains at most r vertices and has a nonnegative weight
w_{e}. Each good j in M also has a nonnegative weight w_{j}. The
value v(S) for a subset S of goods is defined as the sum of the weights of the
goods and edges entirely contained in S.

We design the following polynomial-time approximation algorithms and truthful mechanisms for welfare maximization with bidders with hypergraph valuations.

- For bidders whose valuations correspond to subgraphs of a known graph that is planar (or more generally, excludes a fixed minor), we give a truthful and (1+ε)-approximate mechanism.
- We give a polynomial-time, r-approximation algorithm for welfare maximization with hypergraph-r valuations. Our algorithm randomly rounds a compact linear programming relaxation of the problem.
- We design a different approximation algorithm and use it to give a
polynomial-time, truthful-in-expectation mechanism that has an approximation
factor of O(log
^{r}m).