New Complexity Classes in Locally Checkable Labeling for Local Computation Algorithms

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 $\Theta(\log^k n)$ and $\tilde \Theta(n^{p/q})$ 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 $\Theta(\log n)$ and $\tilde \Theta(n^{1/k})$ in the $\mathrm{VOLUME}$ model constructed by Rosenbaum and Suomela (2020).