Online Resource Allocation via Static Bundle Pricing

TL;DR

This paper develops static bundle pricing mechanisms for online resource allocation with complementarities, achieving performance guarantees that improve exponentially with item multiplicity, and establishes fundamental lower bounds for such problems.

Contribution

It introduces a unified framework for static bundle pricing in online resource allocation with complementarities, providing mechanisms with exponential improvements and matching lower bounds.

Findings

Achieves $O(d^{1/B})$-competitive mechanism for $d$-single-minded settings.

Provides $O(m^{1/(B+1)})$-competitive mechanisms for general cases.

Establishes lower bounds showing no online algorithm can do better than $ ilde{ ext{o}}(m^{1/(B+2)})$.

Abstract

Online Resource Allocation addresses the problem of efficiently allocating limited resources to buyers with incomplete knowledge of future requests. In our setting, buyers arrive sequentially requesting a set of items, each with a value drawn from a known distribution. We study the efficiency of static and anonymous bundle pricing in environments where the buyers' valuations exhibit strong complementarities. In such settings, standard item pricing fails to leverage item multiplicities, while static bundle pricing mechanisms are only known for very restricted domains and their analysis relies on domain-specific arguments. We develop a unified bundle pricing framework for online resource allocation in three well-studied domains with complementarities: (i) single-minded combinatorial auctions with maximum bundle size $d$; (ii) general single-minded combinatorial auctions; and (iii) network routing, where each buyer aims to route a unit of flow from a source node $s$ to a target node $t$ in a capacitated network. Our approach yields static and anonymous bundle pricing mechanisms whose performance improves exponentially with item multiplicity. For the $d$-single-minded setting with minimum item multiplicity $B$, we obtain an $O(d^{1/B})$-competitive mechanism. For general single-minded combinatorial auctions and online network routing, we obtain $O(m^{1/(B+1)})$-competitive mechanisms, where $m$ is the number of items. We complement these results with information-theoretic lower bounds. We show that no online algorithm can achieve a competitive ratio better than $ \widetilde{\Omega}(m^{1/(B+2)})$ for single-minded combinatorial auctions and $ \widetilde{\Omega}(d^{1/(B+1)})$ for the $d$-single-minded setting. Our constructions exploit a deep connection to the extremal combinatorics problem of determining the maximum number of qualitatively independent partitions of a ground set.