  We consider a market where a set of objects is sold to a set of buyers, each
equipped with a valuation function for the objects. The goal of the auctioneer
is to determine reasonable prices together with a stable allocation. One
definition of "reasonable" and "stable" is a Walrasian equilibrium, which is a
tuple consisting of a price vector together with an allocation satisfying the
following desirable properties: (i) the allocation is market-clearing in the
sense that as much as possible is sold, and (ii) the allocation is stable in
the sense that every buyer ends up with an optimal set with respect to the
given prices. Moreover, "buyer-optimal" means that the prices are smallest
possible among all Walrasian prices.
  In this paper, we present a combinatorial network flow algorithm to compute
buyer-optimal Walrasian prices in a multi-unit matching market with additive
valuation functions and buyer demands. The algorithm can be seen as a
generalization of the classical housing market auction and mimics the very
natural procedure of an ascending auction. We use our structural insights to
prove monotonicity of the buyer-optimal Walrasian prices with respect to
changes in supply or demand.
