Faster Dynamic Auctions via Polymatroid Sum

TL;DR

This paper introduces a faster algorithm for dynamic auctions with gross substitutes valuations by leveraging polymatroid sum problems, improving computational efficiency and analyzing price monotonicity properties.

Contribution

It presents a novel push-relabel algorithm for submodular function minimization in auction settings, improving previous running times and establishing price monotonicity results.

Findings

Improved algorithm for finding overdemanded and underdemanded sets.

Monotonicity of Walrasian prices with respect to supply and demand changes.

Equivalence of packing and covering prices with Walrasian prices.

Abstract

We consider dynamic auctions for finding Walrasian equilibria in markets with indivisible items and strong gross substitutes valuation functions. Each price adjustment step in these auction algorithms requires finding an inclusion-wise minimal maximal overdemanded set or an inclusion-wise minimal maximal underdemanded set at the current prices. Both 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 required sets. 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. We further show the following monotonicty properties of Walrasian prices: both the minimal and maximal Walrasian prices can only increase if supply of goods decreases, or if the demand of buyers increases. This is derived from a fine-grained analysis of market prices. We call "packing prices" a price vector such that there is a feasible allocation where each buyer obtains a utility maximizing set. Conversely, by "covering prices" we mean a price vector such that there exists a collection of utility maximizing sets of the buyers that include all available goods. We show that for strong gross substitutes valuations, the component-wise minimal packing prices coincide with the minimal Walrasian prices and the component-wise maximal covering prices coincide with the maximal Walrasian prices. These properties in turn lead to the price monotonicity results.