Efficient Algorithms for Online Subadditive Combinatorial Allocations

2026-10-06

AI Summary

This study addresses the bottleneck in online subadditive combinatorial allocation problems, where constant-competitive algorithms have been established only through existential proofs lacking efficient implementations. To overcome this limitation, this work proposes the first polynomial-time $(6+\varepsilon)$-competitive online algorithm. Methodologically, a recursive beam scoring generator (BSG) is designed to replace the existential arguments underlying randomized scoring. This construction is further integrated with configuration LP relaxations, stochastic dominance properties, and expected polynomial-time algorithms. The proposed approach not only bridges the gap between theoretical computability and practical efficiency but also recovers the offline $(2+\varepsilon)$-approximation result. Furthermore, under identical distribution assumptions, the competitive ratio is improved to $(60/11+\varepsilon)$.

Abstract

For the online combinatorial allocation problem with subadditive valuations, Correa and Cristi (STOC 2023) proved the existence of a $6$-competitive online algorithm, improving on the previous best $O(\log\!\log m)$-competitive online algorithm due to Dütting, Kesselheim, and Lucier (FOCS 2020), where $m$ is the number of items. However, Correa and Cristi's result is existential, and it was left open whether a constant competitive ratio is attainable via an efficient online algorithm that uses a polynomial number of demand oracle queries. In this work, we answer this affirmatively, giving an expected-polynomial-time $(6 + ε)$-competitive online algorithm for any constant $ε> 0$. Our techniques also recover, in the offline setting, the $(2 + ε)$-approximation result of Feige (STOC, 2006). Finally, when the buyers' valuations are drawn from identical distributions, we exploit symmetry to obtain an improved $(60/11+ε)$-competitive algorithm. Starting with the natural configuration LP relaxation for the problem, our main technical contribution is a recursive Bundle Score Generator (BSG) that resolves item conflicts by assigning correlated scores to the items requested by each buyer. Unlike the Random Score Generator whose existence is shown by Correa and Cristi, our BSG is efficiently computable in expected polynomial time. Moreover, it satisfies a stochastic dominance property that is sufficient to recover both Feige's offline result and Correa and Cristi's online result.

Problem

• online combinatorial allocation
• subadditive valuations
• competitive ratio
• demand oracle queries
• efficient algorithms

Innovation

• Online Combinatorial Allocation
• Subadditive Valuations
• Bundle Score Generator
• Competitive Algorithm
• Configuration LP