Generalized Hypergraph Matching via Iterated Packing and Local Ratio

In $k$-hypergraph matching, we are given a collection of sets of size at most $k$, each with an associated weight, and we seek a maximum-weight subcollection whose sets are pairwise disjoint. More generally, in $k$-hypergraph $b$-matching, instead of disjointness we require that every element appears in at most $b$ sets of the subcollection. Our main result is a linear-programming based $(k-1+\tfrac{1}{k})$-approximation algorithm for $k$-hypergraph $b$-matching. This settles the integrality gap when $k$ is one more than a prime power, since it matches a previously-known lower bound. When the hypergraph is bipartite, we are able to improve the approximation ratio to $k-1$, which is also best possible relative to the natural LP. These results are obtained using a more careful application of the \emph{iterated packing} method. Using the bipartite algorithmic integrality gap upper bound, we show that for the family of combinatorial auctions in which anyone can win at most $t$ items, there is a truthful-in-expectation polynomial-time auction that $t$-approximately maximizes social welfare. We also show that our results directly imply new approximations for a generalization of the recently introduced bounded-color matching problem. We also consider the generalization of $b$-matching to \emph{demand matching}, where edges have nonuniform demand values. The best known approximation algorithm for this problem has ratio $2k$ on $k$-hypergraphs. We give a new algorithm, based on local ratio, that obtains the same approximation ratio in a much simpler way.