局部可检查标记中局部计算算法的新复杂度类
New Complexity Classes in Locally Checkable Labeling for Local Computation Algorithms
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中文总结 AI 辅助
本文聚焦局部计算算法，通过复杂度分类理解LCL问题，在VOLUME模型中给出新的LCL复杂度构造并推广到LCAs，展示了特定探针复杂度的LCLs，采用堆叠实例的方法，区别于相关工作。
中文摘要 AI 辅助
局部计算算法（LCAs）是一种特殊的亚线性算法，在对可能海量输入进行探测访问时，需提供对一致解的查询访问且不同查询间不保持状态。本文通过复杂度分类来理解LCA，聚焦局部可检查标记（LCL）问题。在VOLUME模型中提供了新的LCL复杂度构造并推广到LCAs。具体而言，对于任意正整数\(k \geq 1\)和有理数\(p/q \in (0,1]\)，展示了在VOLUME和LCA模型中探针复杂度分别为\(\Theta(\log^k n)\)和\(\tilde \Theta(n^{p/q})\)的LCLs。方法是在Rosenbaum和Suomela（2020）构建的VOLUME模型中堆叠复杂度为\(\Theta(\log n)\)和\(\tilde \Theta(n^{1/k})\)的实例，与Balliu等人（2018）在分布式LOCAL模型中获得类似结果的方法完全不同。
英文摘要
Local Computation Algorithms (LCAs), introduced by Rubinfeld, Tamir, Vardi, and Xie (2011), are a special type of sublinear algorithms that, given probing access to a possibly massive input, are required to provide query access to a consistent solution, without maintaining a state between different queries. In this paper, we try to understand LCA through the lens of complexity classifications, described by the following question: Given a target complexity function $f(n)$, is there a problem whose local computation complexity is $f(n)$, up to polylogarithmic factors? We restrict our focus to Locally Checkable Labeling (LCL) problems, which can be seen as constant-degree constraint satisfaction problems. Possible complexity classes of this problem family have been extensively studied in various distributed computation models, including the $\mathrm{VOLUME}$ model proposed by Rosenbaum and Suomela (2020), which is an invariant of local computation algorithms with additional locality requirements. In this paper, we provide new LCL complexity constructions in the $\mathrm{VOLUME}$ model, and generalize the results to LCAs. Specifically, we show that there are LCLs whose probe complexities in the $\mathrm{VOLUME}$ and LCA models are $Θ(\log^k n)$ and $\tilde Θ(n^{p/q})$ for any positive integer $k \ge 1$ and rational $p/q \in (0,1]$. Our approach, completely different from the approach to a similar result in the distributed $\mathrm{LOCAL}$ model by Balliu et al. (2018), is to stack instances of complexity $Θ(\log n)$ and $\tilde Θ(n^{1/k})$ in the $\mathrm{VOLUME}$ model constructed by Rosenbaum and Suomela (2020).
