[论文解读] On the Complexity of Algorithms with Predictions for Dynamic Graph Problems

Monika Henzinger, Barna Saha+2 more|arXiv (Cornell University)|Jul 31, 2023

Optimization and Search Problems被引用 3

一句话总结

本文提出了动态图算法的预测模型并进行分析，提出三种预测类型——ε-准确、列表准确和延迟有界预测，并基于OMv猜想建立了条件下界。研究发现，局部可校正问题在列表准确预测下除非预测完美，否则仍保持强下界；而局部可约化问题在延迟有界预测下，其运行时间从在线到离线设置间呈现平滑过渡，且在子图连通性和传递闭包等动态图问题上展示了紧致的上界。

ABSTRACT

Algorithms with predictions is a new research direction that leverages machine learned predictions for algorithm design. So far a plethora of recent works have incorporated predictions to improve on worst-case bounds for online problems. In this paper, we initiate the study of complexity of dynamic data structures with predictions, including dynamic graph algorithms. Unlike online algorithms, the goal in dynamic data structures is to maintain the solution efficiently with every update. We investigate three natural models of prediction: (1) δ-accurate predictions where each predicted request matches the true request with probability δ, (2) list-accurate predictions where a true request comes from a list of possible requests, and (3) bounded delay predictions where the true requests are a permutation of the predicted requests. We give general reductions among the prediction models, showing that bounded delay is the strongest prediction model, followed by list-accurate, and δ-accurate. Further, we identify two broad problem classes based on lower bounds due to the Online Matrix Vector (OMv) conjecture. Specifically, we show that locally correctable dynamic problems have strong conditional lower bounds for list-accurate predictions that are equivalent to the non-prediction setting, unless list-accurate predictions are perfect. Moreover, we show that locally reducible dynamic problems have time complexity that degrades gracefully with the quality of bounded delay predictions. We categorize problems with known OMv lower bounds accordingly and give several upper bounds in the delay model that show that our lower bounds are almost tight. We note that concurrent work by v.d.Brand et al. [SODA '24] and Liu and Srinivas [arXiv:2307.08890] independently study dynamic graph algorithms with predictions, but their work is mostly focused on showing upper bounds.

研究动机与目标

• 形式化并分析三种不同的预测模型——ε-准确、列表准确和延迟有界预测，用于动态图算法。
• 基于在线矩阵向量（OMv）猜想，为这些预测模型下的动态问题建立条件性下界。
• 根据问题的局部正确性与可约化特性，刻画动态图问题与预测质量之间的关系。
• 为关键动态图问题在延迟有界预测下提供紧致上界，表明下界近乎最优。
• 通过展示局部可约化问题在延迟有界预测下运行时间从在线到离线设置的平滑过渡，弥合在线与离线设置之间的差距。

提出的方法

• 引入三种预测模型：ε-准确（每个预测正确的概率≥ε）、列表准确（真实请求来自候选列表）、延迟有界（真实请求是预测的排列）。
• 建立模型间的归约关系：延迟有界预测蕴含列表准确预测，后者又蕴含ε-准确预测，从而实现下界在不同模型间的传递。
• 应用OMv猜想将动态问题分类为局部可校正与局部可约化两类，并推导出条件性下界。
• 为延迟有界预测下的#s-△问题提出一种数据结构，其更新时间保持O(1)，查询时间为˜O(d²)，其中d为延迟。
• 使用二叉堆和辅助数组（如D(−)R,d、D(+)R,d、BR,d、B(>)R,d）追踪行与列的更新，并在延迟预测下维持正确性。
• 采用一种动态查询算法，结合预测的矩阵值与实时行/列计数器进行误差校正，以准确计算最大值。

实验结果

研究问题

• RQ1 不同预测模型——ε-准确、列表准确和延迟有界——如何影响动态图算法的复杂度？
• RQ2 基于OMv猜想的条件性下界在带有预测的动态问题中能多大程度上延续适用？
• RQ3 局部正确性/可约化与算法在预测不完美时的鲁棒性之间有何关系？
• RQ4 能否在延迟有界预测下为动态图问题实现紧致上界？其与最坏情况下的界相比如何？
• RQ5 当预测存在延迟时，算法性能是否在从在线到离线设置间呈现平滑过渡？

主要发现

• 对于局部可校正的动态问题，除非预测完美，否则列表准确预测不会使条件性下界优于无预测的情形。
• 对于局部可约化问题，延迟有界预测使运行时间从在线到离线设置间实现平滑过渡，查询时间随延迟d呈˜O(d²)增长。
• #s-△问题在延迟有界预测下实现˜O(1)更新时间与˜O(d²)查询时间，与条件性下界仅相差对数因子。
• Erickson最大值问题的算法在延迟有界预测下维持˜O(1)更新时间与˜O(d²)查询时间，表明其近乎最优。
• 子图连通性问题的数据结构在延迟有界预测下支持˜O(1)更新与˜O(d²)查询，通过动态追踪行与列计数器维持正确性。
• 预测模型间的归约关系表明，延迟有界是最强的模型，其下界可推导至列表准确与ε-准确预测。