Online Resource Allocation via Static Bundle Pricing

TL;DR

This work develops a unified, LP-guided framework for online resource allocation with strong complementarities under known valuation distributions. By solving a scaled ex-ante LP and posting a static, anonymous bundle menu, the authors transform fractional welfare into practical posted prices, backed by a rigorous unconstrained benchmark and load-concentration analysis. They obtain strong, capacity-amplified guarantees: $O(d^{1/B})$ for $d$-single-minded and $O(m^{1/(B+1)})$ for general single-minded and graph-routing settings, complemented by information-theoretic lower bounds that match the dependence on $m$, $d$, and $B$ up to constants. The results reveal a structural transition where modest increases in item multiplicity exponentially improve performance, and they connect the lower bounds to extremal combinatorics through qualitatively independent partitions. The framework also covers a broad spectrum of settings and paves the way for future work on sample-based implementations and tighter bounds.

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 demanding a set of items, each with a value drawn from a known distribution. We study environments where buyers' valuations exhibit complementarities. In such settings, standard item-pricing mechanisms fail to leverage item multiplicities, while existing static bundle-pricing mechanisms rely on problem-specific arguments that do not generalize. We develop a unified technique for online resource allocation with complementarities for three domains: (i) single-minded combinatorial auctions with maximum bundle size $d$, (ii) general single-minded combinatorial auctions, and (iii) a graph-based routing model in which buyers request to route a unit of flow from a source node $s$ to a target node $t$ in a capacitated graph. Our approach yields static and anonymous bundle-pricing mechanisms whose performance improves exponentially with item capacities. For the $d$-single-minded setting with minimum item capacity $B$, we obtain an $O(d^{1/B})$-competitive mechanism, recovering the known $O(d)$ bound for unit capacities ($B=1$) and achieving exponentially better guarantees as capacities grow. For general single-minded combinatorial auctions and the graph-routing model, 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 $Ω((m/\ln m)^{1/(B+2)})$ in the general single-minded setting and $Ω((d/\ln d)^{1/(B+1)})$ in the $d$-single-minded setting. In doing so, we reveal a deep connection to the extremal combinatorics problem of determining the maximum number of qualitatively independent partitions of a ground set.

Theorems & Definitions (58)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1
• Definition 2.2
• Lemma 2.3