Local Computation Algorithms (LCA), as introduced by Rubinfeld, Tamir, Vardi, and Xie (2011), are a type of ultra-efficient algorithms which, given access to a (large) input for a given computational task, are required to provide fast query access to a consistent output solution, without maintaining a state between queries. This paradigm of computation in particular allows for hugely distributed algorithms, where independent instances of a given LCA provide consistent access to a common output solution. The past decade has seen a significant amount of work on LCAs, by and large focusing on graph problems. In this paper, we initiate the study of Local Computation Algorithms for perhaps the archetypal combinatorial optimization problem, Knapsack. We first establish strong impossibility results, ruling out the existence of any non-trivial LCA for Knapsack as several of its relaxations. We then show how equipping the LCA with additional access to the Knapsack instance, namely, weighted item sampling, allows one to circumvent these impossibility results, and obtain sublinear-time and query LCAs. Our positive result draws on a connection to the recent notion of reproducibility for learning algorithms (Impagliazzo, Lei, Pitassi, and Sorrell, 2022), a connection we believe to be of independent interest for the design of LCAs.

翻译：局部计算算法（Local Computation Algorithms，LCA）由Rubinfeld、Tamir、Vardi和Xie（2011年）提出，是一类超高效算法。该类算法在获得对（大规模）计算任务输入数据的访问权限后，无需在查询间保持状态，即能快速提供对一致性输出解的查询访问。这种计算范式特别适用于高度分布式算法场景，其中独立运行的LCA实例可对同一输出解提供一致性访问。过去十年间，针对LCA的研究取得了丰硕成果，其中绝大多数聚焦于图论问题。本文首次针对组合优化中的典型问题——背包问题——展开局部计算算法的系统性研究。我们首先建立了强不可能性结果，证明背包问题及其若干松弛形式均不存在任何非平凡LCA。随后我们证明，通过为LCA提供对背包问题实例的额外访问机制（即加权项采样），可以规避这些不可能性结果，从而获得亚线性时间与查询复杂度的LCA。我们的正向结果借鉴了近期机器学习算法可重现性概念（Impagliazzo、Lei、Pitassi和Sorrell，2022年）的相关研究，我们认为这种关联性对于LCA的设计具有独立的研究价值。