Gaps and Augmentations in Bayesian Scheduling

2026-09-27

AI Summary

This study investigates the impact of prior information, Bayesian incentive compatibility (BIC), and resource augmentation on approximation ratios in unrelated machine scheduling. Through randomized algorithm design and resource augmentation analysis, it rigorously proves that randomized BIC mechanisms outperform deterministic ones in two-machine settings, identifies an 8/7 integrality gap, and proposes a collision-parameter-based resource augmentation approximation scheme. The results establish a 4/3 asymptotic lower bound and improve the prior-free randomized approximation ratio to 7/4. Furthermore, by leveraging sampling-level augmentation, the work achieves a (1+ε)-approximation, providing theoretical foundations for efficient scheduling under specific conditions.

Abstract

The recent resolution of the Nisan--Ronen conjecture~\cite{NR,CKK} establishes that the optimal worst-case approximation ratio of deterministic truthful mechanisms for makespan on m unrelated machines is exactly m. We ask how prior information, Bayesian incentive compatibility (BIC), randomization, and machine-side resource augmentation change this barrier. We obtain three main results. First, we prove an asymptotic 4/3 lower bound for randomized BIC mechanisms with two machines, strengthening the previous 1.2 deterministic-BIC lower bound~\cite{MS}. Second, in the prior-free setting for two machines, randomization improves the known truthful ratio from 2 to 7/4~\cite{NR}. We show that randomized BIC scheduling mechanisms are likewise strictly more powerful than their deterministic counterparts, exhibiting an asymptotic BIC-integrality gap of 8/7. Thus, optimality in prior-dependent BIC scheduling can strictly require randomization. This parallels the role of lotteries in multidimensional revenue maximization, where randomization can strictly improve revenue~\cite{MV,BCKW}. Third, we show that when job assignments are sufficiently well spread across machines (more formally, when the pairwise collision parameter $\Delta_r$ is bounded by a constant independent of both the number of machines m and the number of sampled layers r), then $O(m/\varepsilon)$ sampled layers suffice for the standard truthful MinWork mechanism to achieve a $1+\varepsilon$-approximation to the original first-best benchmark. Equivalently, $O(m/\varepsilon)$ sampled replicas per machine suffice. This resource-augmentation result parallels the Bulow--Klemperer perspective~\cite{BK,EFFTW}. As a by-product, in the standard unaugmented model, MinWork achieves a $(1+\frac{\Delta_1(m-1)}{2})$-approximation to the first-best benchmark for every prior.

Problem

• Bayesian scheduling
• makespan minimization
• approximation ratio
• randomized mechanisms
• resource augmentation

Innovation

• Bayesian incentive compatibility
• randomized mechanisms
• resource augmentation
• makespan scheduling
• approximation ratio