Faster Ascending Auctions via Polymatroid Sum

We consider ascending auctions for finding Walrasian equilibria in markets with indivisible items and gross substitutes valuation functions. Each price increase step in the auction algorithm requires finding an inclusion-wise minimal maximal overdemanded set at the current prices. This can be formulated as a submodular function minimization problem. We observe that minimizing this submodular function corresponds to a polymatroid sum problem, and using this viewpoint, we give a fast and simple push-relabel algorithm for finding the minimal maximal overdemanded set. This improves on the previously best running time of Murota, Shioura and Yang (ISAAC 2013). Our algorithm is an adaptation of the push-relabel framework by Frank and Mikl\'os (JJIAM, 2012) to the particular setting. We obtain a further improvement for the special case of unit-supplies. Furthermore, we show that for gross substitutes valuations, the component-wise minimal competitive prices are the same as the minimal Walrasian prices. This enables us to derive monotonicity properties of the Walrasian prices. Namely, we show that the minimal Walrasian prices can only increase if supply decreases, or demand increases.